<PAGE>

                                                                    EXHIBIT 4.10





                              GUARANTEE AGREEMENT

                              GREATER BAY BANCORP

                           Dated as of July 16, 2001
<PAGE>

                               TABLE OF CONTENTS


<TABLE>
<CAPTION>
                                                                      Page
<S>          <C>                                                        <C>

                                   ARTICLE I
                        DEFINITIONS AND INTERPRETATION


SECTION 1.1  Definitions and Interpretation............................  1

                                  ARTICLE II
                POWERS, DUTIES AND RIGHTS OF GUARANTEE TRUSTEE

SECTION 2.1  Powers and Duties of the Guarantee Trustee................  4
SECTION 2.2  Certain Rights of Guarantee Trustee.......................  5
SECTION 2.3  Not Responsible for Recitals or Issuance of Guarantee.....  7
SECTION 2.4  Events of Default; Waiver.................................  7
SECTION 2.5  Events of Default; Notice.................................  7

                                  ARTICLE III
                               GUARANTEE TRUSTEE

SECTION 3.1  Guarantee Trustee; Eligibility............................  8

SECTION 3.2  Appointment, Removal and Resignation of Guarantee Trustee.  9

                                  ARTICLE IV
                                   GUARANTEE

SECTION 4.1  Guarantee.................................................  9
SECTION 4.2  Waiver of Notice and Demand............................... 10
SECTION 4.3  Obligations Not Affected.................................. 10
SECTION 4.4  Rights of Holders......................................... 11
SECTION 4.5  Guarantee of Payment...................................... 11
SECTION 4.6  Subrogation............................................... 11
SECTION 4.7  Independent Obligations................................... 12
SECTION 4.8  Enforcement............................................... 12

                                   ARTICLE V
                   LIMITATION OF TRANSACTIONS; SUBORDINATION

SECTION 5.1  Limitation of Transactions................................ 12

SECTION 5.2  Ranking................................................... 13
</TABLE>

                                       i
<PAGE>

                                  ARTICLE VI
                                  TERMINATION
<TABLE>
<S>          <C>                                                       <C>
SECTION 6.1  Termination............................................... 13

                                  ARTICLE VII
                                INDEMNIFICATION


SECTION 7.1  Exculpation............................................... 13
SECTION 7.2  Indemnification........................................... 14
SECTION 7.3  Compensation; Reimbursement of Expenses................... 15

                                 ARTICLE VIII
                                 MISCELLANEOUS

SECTION 8.1  Successors and Assigns.................................... 15
SECTION 8.2  Amendments................................................ 16
SECTION 8.3  Notices................................................... 16
SECTION 8.4  Benefit................................................... 16
SECTION 8.5  Governing Law............................................. 17
SECTION 8.6  Counterparts.............................................. 17
</TABLE>

                                      ii
<PAGE>

                              GUARANTEE AGREEMENT
                              -------------------

          This GUARANTEE AGREEMENT (the "Guarantee"), dated as of July 16, 2001,
is executed and delivered by Greater Bay Bancorp, a bank holding company
incorporated in California (the "Guarantor"), and The Bank of New York, as
trustee (the "Guarantee Trustee"), for the benefit of the Holders (as defined
herein) from time to time of the Capital Securities (as defined herein) of GBB
Capital VI, a Delaware statutory business trust (the "Issuer").

          WHEREAS, pursuant to an Amended and Restated Declaration of Trust (the
"Declaration"), dated as of July 16, 2001, among the trustees named therein of
the Issuer, Greater Bay Bancorp, as sponsor, and the Holders from time to time
of undivided beneficial interests in the assets of the Issuer, the Issuer is
issuing on the date hereof securities, having an aggregate liquidation amount of
up to $15,000,000, designated the Floating Rate MMCapS(SM) (the "Capital
Securities");

          WHEREAS, as incentive for the Holders to purchase the Capital
Securities, the Guarantor desires irrevocably and unconditionally to agree, to
the extent set forth in this Guarantee, to pay to the Holders of Capital
Securities the Guarantee Payments (as defined herein) and to make certain other
payments on the terms and conditions set forth herein; and

          NOW, THEREFORE, in consideration of the purchase by each Holder of the
Capital Securities, which purchase the Guarantor hereby agrees shall benefit the
Guarantor, the Guarantor executes and delivers this Guarantee for the benefit of
the Holders.

                                   ARTICLE I
                         DEFINITIONS AND INTERPRETATION

SECTION 1.1  Definitions and Interpretation
             ------------------------------

In this Guarantee, unless the context otherwise requires:

               (a)  capitalized terms used in this Guarantee but not defined in
     the preamble above have the respective meanings assigned to them in this
     Section 1.1;

               (b)  a term defined anywhere in this Guarantee has the same
     meaning throughout;

               (c)  all references to "the Guarantee" or "this Guarantee" are to
     this Guarantee as modified, supplemented or amended from time to time;

               (d)  all references in this Guarantee to Articles and Sections
     are to Articles and Sections of this Guarantee, unless otherwise specified;

               (e)  terms defined in the Declaration as at the date of execution
     of this Guarantee have the same meanings when used in this Guarantee,
     unless otherwise defined in this Guarantee or unless the context otherwise
     requires; and

               (f)  a reference to the singular includes the plural and vice
     versa.
<PAGE>

          "Beneficiaries" means any Person to whom the Issuer is or hereafter
becomes indebted or liable.

          "Corporate Trust Office" means the office of the Guarantee Trustee at
which the corporate trust business of the Guarantee Trustee shall, at any
particular time, be principally administered, which office at the date of
execution of this Guarantee Agreement is located at 101 Barclay Street, Floor
21W, New York, NY 10286.

          "Covered Person" means any Holder of Capital Securities.

          "Debentures" means the junior subordinated debentures of Greater Bay
Bancorp, designated the Floating Rate Junior Subordinated Deferrable Interest
Debentures due 2031, held by the Institutional Trustee (as defined in the
Declaration) of the Issuer.

          "Event of Default" has the meaning set forth in Section 2.4.

          "Guarantee Payments" means the following payments or distributions,
without duplication, with respect to the Capital Securities, to the extent not
paid or made by the Issuer: (i) any accrued and unpaid Distributions (as defined
in the Declaration) which are required to be paid on such Capital Securities to
the extent the Issuer shall have funds available in the Property Account (as
defined in the Declaration) therefor at such time, (ii) the Redemption Price (as
defined in the Indenture) to the extent the Issuer has funds available in the
Property Account therefor at such time, with respect to any Capital Securities
called for redemption by the Issuer, (iii) the Special Redemption Price (as
defined in the Indenture) to the extent the Issuer has funds available in the
Property Account therefor at such time, with respect to Capital Securities
called for redemption upon the occurrence of a Special Event (as defined in the
Indenture), and (iv) upon a voluntary or involuntary liquidation, dissolution,
winding-up or termination of the Issuer (other than in connection with the
distribution of Debentures to the Holders of the Capital Securities in exchange
therefor as provided in the Declaration), the lesser of (a) the aggregate of the
liquidation amount and all accrued and unpaid Distributions on the Capital
Securities to the date of payment, to the extent the Issuer shall have funds
available in the Property Account therefor at such time, and (b) the amount of
assets of the Issuer remaining available for distribution to Holders in
liquidation of the Issuer after satisfaction of liabilities to creditors of the
Issuer as required by applicable law (in either case, the "Liquidation
Distribution").

          "Guarantee Trustee" means The Bank of New York , until a Successor
Guarantee Trustee has been appointed and has accepted such appointment pursuant
to the terms of this Guarantee and thereafter means each such Successor
Guarantee Trustee.

          "Holder" means any holder, as registered on the books and records of
the Issuer, of any Capital Securities; provided, however, that, in determining
                                       --------  -------
whether the holders of the requisite percentage of Capital Securities have given
any request, notice, consent or waiver hereunder, "Holder" shall not include the
Guarantor or any Affiliate of the Guarantor.

          "Indemnified Person" means the Guarantee Trustee, any Affiliate of the
Guarantee Trustee, or any officers, directors, shareholders, members, partners,
employees, representatives, nominees, custodians or agents of the Guarantee
Trustee.

                                       2
<PAGE>

          "Indenture" means the Indenture dated as of July 16, 2001, between the
Guarantor and The Bank of New York, not in its individual capacity but solely as
trustee, and any indenture supplemental thereto pursuant to which the Debentures
are to be issued to the Institutional Trustee of the Issuer.

          "Liquidation Distribution" has the meaning set forth in the definition
of "Guarantee Payments" herein.

          "Majority in liquidation amount of the Capital Securities" means
Holder(s) of outstanding Capital Securities, voting together as a class, but
separately from the holders of Common Securities, of more than 50% of the
aggregate liquidation amount (including the stated amount that would be paid on
redemption, liquidation or otherwise, plus accrued and unpaid Distributions to,
but excluding, the date upon which the voting percentages are determined) of all
Capital Securities then outstanding.

          "Obligations" means any costs, expenses or liabilities (but not
including liabilities related to taxes) of the Issuer, other than obligations of
the Issuer to pay to holders of any Trust Securities the amounts due such
holders pursuant to the terms of the Trust Securities.

          "Officer's Certificate" means, with respect to any Person, a
certificate signed by one Authorized Officer of such Person.  Any Officer's
Certificate delivered with respect to compliance with a condition or covenant
provided for in this Guarantee shall include:

               (a)  a statement that each officer signing the Officer's
     Certificate has read the covenant or condition and the definitions relating
     thereto;

               (b)  a brief statement of the nature and scope of the examination
     or investigation undertaken by each officer in rendering the Officer's
     Certificate;

               (c)  a statement that each such officer has made such examination
     or investigation as, in such officer's opinion, is necessary to enable such
     officer to express an informed opinion as to whether or not such covenant
     or condition has been complied with; and

               (d)  a statement as to whether, in the opinion of each such
     officer, such condition or covenant has been complied with.

          "Person" means a legal person, including any individual, corporation,
estate, partnership, joint venture, association, joint stock company, limited
liability company, trust, unincorporated association, or government or any
agency or political subdivision thereof, or any other entity of whatever nature.

          "Responsible Officer" means, with respect to the Guarantee Trustee,
any officer within the Corporate Trust Office of the Guarantee Trustee including
any Vice President, Assistant Vice President, Secretary, Assistant Secretary or
any other officer of the Guarantee Trustee customarily performing functions
similar to those performed by any of the above designated officers and also,
with respect to a particular corporate trust matter, any other officer to whom
such matter is referred because of that officer's knowledge of and familiarity
with the particular subject.

                                       3
<PAGE>

          "Successor Guarantee Trustee" means a successor Guarantee Trustee
possessing the qualifications to act as Guarantee Trustee under Section 3.1.

          "Trust Securities" means the Common Securities and the Capital
Securities.

                                   ARTICLE II
                 POWERS, DUTIES AND RIGHTS OF GUARANTEE TRUSTEE

            SECTION 2.1  Powers and Duties of the Guarantee Trustee
                         ------------------------------------------

               (a)  This Guarantee shall be held by the Guarantee Trustee for
     the benefit of the Holders of the Capital Securities, and the Guarantee
     Trustee shall not transfer this Guarantee to any Person except a Holder of
     Capital Securities exercising his or her rights pursuant to Section 4.4 (b)
     or to a Successor Guarantee Trustee on acceptance by such Successor
     Guarantee Trustee of its appointment to act as Successor Guarantee Trustee.
     The right, title and interest of the Guarantee Trustee shall automatically
     vest in any Successor Guarantee Trustee, and such vesting and cessation of
     title shall be effective whether or not conveyancing documents have been
     executed and delivered pursuant to the appointment of such Successor
     Guarantee Trustee.

               (b)  If an Event of Default actually known to a Responsible
     Officer of the Guarantee Trustee has occurred and is continuing, the
     Guarantee Trustee shall enforce this Guarantee for the benefit of the
     Holders of the Capital Securities.

               (c)  The Guarantee Trustee, before the occurrence of any Event of
     Default and after curing all Events of Default that may have occurred,
     shall undertake to perform only such duties as are specifically set forth
     in this Guarantee, and no implied covenants shall be read into this
     Guarantee against the Guarantee Trustee.  In case an Event of Default has
     occurred (that has not been cured or waived pursuant to Section 2.4) and is
     actually known to a Responsible Officer of the Guarantee Trustee, the
     Guarantee Trustee shall exercise such of the rights and powers vested in it
     by this Guarantee, and use the same degree of care and skill in its
     exercise thereof, as a prudent person would exercise or use under the
     circumstances in the conduct of his or her own affairs.

               (d)  No provision of this Guarantee shall be construed to relieve
     the Guarantee Trustee from liability for its own negligent action, its own
     negligent failure to act, or its own willful misconduct, except that:

                    (i)  prior to the occurrence of any Event of Default and
          after the curing or waiving of all such Events of Default that may
          have occurred:

                         (A)  the duties and obligations of the Guarantee
               Trustee shall be determined solely by the express provisions of
               this Guarantee, and the Guarantee Trustee shall not be liable
               except for the performance of such duties and obligations as are
               specifically set forth in this Guarantee, and no implied
               covenants or obligations shall be read into this Guarantee
               against the Guarantee Trustee; and

                                       4
<PAGE>

                         (B)  in the absence of bad faith on the part of the
               Guarantee Trustee, the Guarantee Trustee may conclusively rely,
               as to the truth of the statements and the correctness of the
               opinions expressed therein, upon any certificates or opinions
               furnished to the Guarantee Trustee and conforming to the
               requirements of this Guarantee; but in the case of any such
               certificates or opinions furnished to the Guarantee Trustee, the
               Guarantee Trustee shall be under a duty to examine the same to
               determine whether or not they conform to the requirements of this
               Guarantee;

                    (ii)  the Guarantee Trustee shall not be liable for any
          error of judgment made in good faith by a Responsible Officer of the
          Guarantee Trustee, unless it shall be proved that such Responsible
          Officer of the Guarantee Trustee or the Guarantee Trustee was
          negligent in ascertaining the pertinent facts upon which such judgment
          was made;

                    (iii)  the Guarantee Trustee shall not be liable with
          respect to any action taken or omitted to be taken by it in good faith
          in accordance with the written direction of the Holders of not less
          than a Majority in liquidation amount of the Capital Securities
          relating to the time, method and place of conducting any proceeding
          for any remedy available to the Guarantee Trustee, or exercising any
          trust or power conferred upon the Guarantee Trustee under this
          Guarantee; and

                    (iv)  no provision of this Guarantee shall require the
          Guarantee Trustee to expend or risk its own funds or otherwise incur
          personal financial liability in the performance of any of its duties
          or in the exercise of any of its rights or powers, if the Guarantee
          Trustee shall have reasonable grounds for believing that the repayment
          of such funds is not reasonably assured to it under the terms of this
          Guarantee, or security and indemnity, reasonably satisfactory to the
          Guarantee Trustee, against such risk or liability is not reasonably
          assured to it.

SECTION 2.2  Certain Rights of Guarantee Trustee
             -----------------------------------

               (a) Subject to the provisions of Section 2.1:

                    (i)  The Guarantee Trustee may conclusively rely, and shall
          be fully protected in acting or refraining from acting upon, any
          resolution, certificate, statement, instrument, opinion, report,
          notice, request, direction, consent, order, bond, debenture, note,
          other evidence of indebtedness or other paper or document believed by
          it to be genuine and to have been signed, sent or presented by the
          proper party or parties.

                    (ii) Any direction or act of the Guarantor contemplated by
          this Guarantee shall be sufficiently evidenced by an Officer's
          Certificate.

                    (iii)  Whenever, in the administration of this Guarantee,
          the Guarantee Trustee shall deem it desirable that a matter be proved
          or established before taking, suffering or omitting any action
          hereunder, the Guarantee Trustee (unless other evidence is herein
          specifically prescribed) may, in the absence of bad faith

                                       5
<PAGE>

          on its part, request and conclusively rely upon an Officer's
          Certificate of the Guarantor which, upon receipt of such request,
          shall be promptly delivered by the Guarantor.

                    (iv)  The Guarantee Trustee shall have no duty to see to any
          recording, filing or registration of any instrument (or any
          rerecording, refiling or registration thereof).

                    (v)  The Guarantee Trustee may consult with counsel of its
          selection, and the advice or opinion of such counsel with respect to
          legal matters shall be full and complete authorization and protection
          in respect of any action taken, suffered or omitted by it hereunder in
          good faith and in accordance with such advice or opinion.  Such
          counsel may be counsel to the Guarantor or any of its Affiliates and
          may include any of its employees.  The Guarantee Trustee shall have
          the right at any time to seek instructions concerning the
          administration of this Guarantee from any court of competent
          jurisdiction.

                    (vi)  The Guarantee Trustee shall be under no obligation to
          exercise any of the rights or powers vested in it by this Guarantee at
          the request or direction of any Holder, unless such Holder shall have
          provided to the Guarantee Trustee such security and indemnity,
          reasonably satisfactory to the Guarantee Trustee, against the costs,
          expenses (including attorneys' fees and expenses and the expenses of
          the Guarantee Trustee's agents, nominees or custodians) and
          liabilities that might be incurred by it in complying with such
          request or direction, including such reasonable advances as may be
          requested by the Guarantee Trustee; provided, however, that nothing
                                              --------  -------
          contained in this Section 2.2(a)(vi) shall be taken to relieve the
          Guarantee Trustee, upon the occurrence of an Event of Default, of its
          obligation to exercise the rights and powers vested in it by this
          Guarantee.

                    (vii)  The Guarantee Trustee shall not be bound to make any
          investigation into the facts or matters stated in any resolution,
          certificate, statement, instrument, opinion, report, notice, request,
          direction, consent, order, bond, debenture, note, other evidence of
          indebtedness or other paper or document, but the Guarantee Trustee, in
          its discretion, may make such further inquiry or investigation into
          such facts or matters as it may see fit.

                    (viii)  The Guarantee Trustee may execute any of the trusts
          or powers hereunder or perform any duties hereunder either directly or
          by or through agents, nominees, custodians or attorneys, and the
          Guarantee Trustee shall not be responsible for any misconduct or
          negligence on the part of any agent or attorney appointed with due
          care by it hereunder.

                    (ix)  Any action taken by the Guarantee Trustee or its
          agents hereunder shall bind the Holders of the Capital Securities, and
          the signature of the Guarantee Trustee or its agents alone shall be
          sufficient and effective to perform any such action.  No third party
          shall be required to inquire as to the authority of the Guarantee
          Trustee to so act or as to its compliance with any of the terms and

                                       6
<PAGE>

          provisions of this Guarantee, both of which shall be conclusively
          evidenced by the Guarantee Trustee's or its agent's taking such
          action.

                    (x)  Whenever in the administration of this Guarantee the
          Guarantee Trustee shall deem it desirable to receive instructions with
          respect to enforcing any remedy or right or taking any other action
          hereunder, the Guarantee Trustee (A) may request instructions from the
          Holders of a Majority in liquidation amount of the Capital Securities,
          (B) may refrain from enforcing such remedy or right or taking such
          other action until such instructions are received and (C) shall be
          protected in conclusively relying on or acting in accordance with such
          instructions.

                    (xi)  The Guarantee Trustee shall not be liable for any
          action taken, suffered, or omitted to be taken by it in good faith and
          reasonably believed by it to be authorized or within the discretion or
          rights or powers conferred upon it by this Guarantee.

               (b)  No provision of this Guarantee shall be deemed to impose any
     duty or obligation on the Guarantee Trustee to perform any act or acts or
     exercise any right, power, duty or obligation conferred or imposed on it,
     in any jurisdiction in which it shall be illegal or in which the Guarantee
     Trustee shall be unqualified or incompetent in accordance with applicable
     law to perform any such act or acts or to exercise any such right, power,
     duty or obligation.  No permissive power or authority available to the
     Guarantee Trustee shall be construed to be a duty.

SECTION 2.3  Not Responsible for Recitals or Issuance of Guarantee
             -----------------------------------------------------

          The recitals contained in this Guarantee shall be taken as the
statements of the Guarantor, and the Guarantee Trustee does not assume any
responsibility for their correctness.  The Guarantee Trustee makes no
representation as to the validity or sufficiency of this Guarantee.

SECTION 2.4  Events of Default; Waiver
             -------------------------

          (a)  An Event of Default under this Guarantee will occur upon the
failure of the Guarantor to perform any of its payment or other obligations
hereunder.

          (b)  The Holders of a Majority in liquidation amount of Capital
Securities may, voting or consenting as a class, on behalf of the Holders of all
of the Capital Securities, waive any past Event of Default and its consequences.
Upon such waiver, any such Event of Default shall cease to exist, and shall be
deemed to have been cured, for every purpose of this Guarantee, but no such
waiver shall extend to any subsequent or other default or Event of Default or
impair any right consequent thereon.

SECTION 2.5  Events of Default; Notice
             -------------------------

               (a)  The Guarantee Trustee shall, within 90 days after the
     occurrence of an Event of Default, transmit by mail, first class postage
     prepaid, to the Holders of the Capital

                                       7
<PAGE>

     Securities, notices of all Events of Default actually known to a
     Responsible Officer of the Guarantee Trustee, unless such defaults have
     been cured before the giving of such notice, provided, however, that the
                                                  --------  -------
     Guarantee Trustee shall be protected in withholding such notice if and so
     long as a Responsible Officer of the Guarantee Trustee in good faith
     determines that the withholding of such notice is in the interests of the
     Holders of the Capital Securities.

               (b)  The Guarantee Trustee shall not be deemed to have knowledge
     of any Event of Default unless the Guarantee Trustee shall have received
     written notice from the Guarantor or a Holder of the Capital Securities
     (except in the case of a payment default), or a Responsible Officer of the
     Guarantee Trustee charged with the administration of this Guarantee shall
     have obtained actual knowledge, thereof.

                                  ARTICLE III
                               GUARANTEE TRUSTEE

SECTION 3.1  Guarantee Trustee; Eligibility
             ------------------------------

          (a)  There shall at all times be a Guarantee Trustee which shall:

               (i)  not be an Affiliate of the Guarantor; and

               (ii)  be a corporation organized and doing business under the
          laws of the United States of America or any State or Territory thereof
          or of the District of Columbia, or Person authorized under such laws
          to exercise corporate trust powers, having a combined capital and
          surplus of at least 50 million U.S. dollars ($50,000,000), and subject
          to supervision or examination by Federal, State, Territorial or
          District of Columbia authority.  If such corporation publishes reports
          of condition at least annually, pursuant to law or to the requirements
          of the supervising or examining authority referred to above, then, for
          the purposes of this Section 3.1(a) (ii), the combined capital and
          surplus of such corporation shall be deemed to be its combined capital
          and surplus as set forth in its most recent report of condition so
          published.

          (b)  If at any time the Guarantee Trustee shall cease to be eligible
     to so act under Section 3.1(a), the Guarantee Trustee shall immediately
     resign in the manner and with the effect set out in Section 3.2(c).

          (c)  If the Guarantee Trustee has or shall acquire any "conflicting
     interest" within the meaning of Section 310(b) of the Trust Indenture Act,
     the Guarantee Trustee shall either eliminate such interest or resign to the
     extent and in the manner provided by, and subject to this Guarantee.

                                       8
<PAGE>

SECTION 3.2  Appointment, Removal and Resignation of Guarantee Trustee
             ---------------------------------------------------------

          (a)  Subject to Section 3.2(b), the Guarantee Trustee may be appointed
     or removed without cause at any time by the Guarantor except during an
     Event of Default.

          (b)  The Guarantee Trustee shall not be removed in accordance with
     Section 3.2(a) until a Successor Guarantee Trustee has been appointed and
     has accepted such appointment by written instrument executed by such
     Successor Guarantee Trustee and delivered to the Guarantor.

          (c)  The Guarantee Trustee appointed to office shall hold office until
     a Successor Guarantee Trustee shall have been appointed or until its
     removal or resignation.  The Guarantee Trustee may resign from office
     (without need for prior or subsequent accounting) by an instrument in
     writing executed by the Guarantee Trustee and delivered to the Guarantor,
     which resignation shall not take effect until a Successor Guarantee Trustee
     has been appointed and has accepted such appointment by an instrument in
     writing executed by such Successor Guarantee Trustee and delivered to the
     Guarantor and the resigning Guarantee Trustee.

          (d)  If no Successor Guarantee Trustee shall have been appointed and
     accepted appointment as provided in this Section 3.2 within 60 days after
     delivery of an instrument of removal or resignation, the Guarantee Trustee
     resigning or being removed may petition any court of competent jurisdiction
     for appointment of a Successor Guarantee Trustee.  Such court may
     thereupon, after prescribing such notice, if any, as it may deem proper,
     appoint a Successor Guarantee Trustee.

          (e)  No Guarantee Trustee shall be liable for the acts or omissions to
     act of any Successor Guarantee Trustee.

          (f)  Upon termination of this Guarantee or removal or resignation of
     the Guarantee Trustee pursuant to this Section 3.2, the Guarantor shall pay
     to the Guarantee Trustee all amounts owing to the Guarantee Trustee under
     Sections 7.2 and 7.3 accrued to the date of such termination, removal or
     resignation.

                                   ARTICLE IV
                                   GUARANTEE

SECTION 4.1  Guarantee
             ---------

          (a)  The Guarantor irrevocably and unconditionally agrees to pay in
full to the Holders the Guarantee Payments (without duplication of amounts
theretofore paid by the Issuer), as and when due, regardless of any defense
(except as defense of payment by the Issuer), right of set-off or counterclaim
that the Issuer may have or assert.  The Guarantor fully, knowingly and
unconditionally waives any right that the Guarantor may have to revoke this
Guarantee under Section 2815 of the California Civil Code or otherwise.  The
Guarantor's obligation to make a Guarantee Payment may be satisfied by direct
payment of the required amounts by the Guarantor to the Holders or by causing
the Issuer to pay such amounts to the Holders.

                                       9
<PAGE>

          (b)  The Guarantor hereby also agrees to assume any and all
Obligations of the Issuer and in the event any such Obligation is not so
assumed, subject to the terms and conditions hereof, the Guarantor hereby
irrevocably and unconditionally guarantees to each Beneficiary the full payment,
when and as due, of any and all obligations to such Beneficiaries.  This
Agreement is intended to be for the Beneficiaries who have received notice
hereof.

SECTION 4.2  Waiver of Notice and Demand
             ---------------------------

          The Guarantor hereby waives notice of acceptance of this Guarantee and
of any liability to which it applies or may apply, presentment, demand for
payment, any right to require a proceeding first against the Issuer or any other
Person before proceeding against the Guarantor, protest, notice of nonpayment,
notice of dishonor, notice of redemption and all other notices and demands.

SECTION 4.3  Obligations Not Affected
             ------------------------

          The obligations, covenants, agreements and duties of the Guarantor
under this Guarantee shall in no way be affected or impaired by reason of the
happening from time to time of any of the following:

          (a)  the release or waiver, by operation of law or otherwise, of the
     performance or observance by the Issuer of any express or implied
     agreement, covenant, term or condition relating to the Capital Securities
     to be performed or observed by the Issuer;

          (b)  the extension of time for the payment by the Issuer of all or any
     portion of the Distributions, Redemption Price, Special Redemption Price,
     Liquidation Distribution or any other sums payable under the terms of the
     Capital Securities or the extension of time for the performance of any
     other obligation under, arising out of, or in connection with, the Capital
     Securities (other than an extension of time for payment of Distributions,
     Redemption Price, Special Redemption Price, Liquidation Distribution or
     other sum payable that results from the extension of any interest payment
     period on the Debentures or any extension of the maturity date of the
     Debentures permitted by the Indenture);

          (c)  any failure, omission, delay or lack of diligence on the part of
     the Holders to enforce, assert or exercise any right, privilege, power or
     remedy conferred on the Holders pursuant to the terms of the Capital
     Securities, or any action on the part of the Issuer granting indulgence or
     extension of any kind;

          (d)  the voluntary or involuntary liquidation, dissolution, sale of
     any collateral, receivership, insolvency, bankruptcy, assignment for the
     benefit of creditors, reorganization, arrangement, composition or
     readjustment of debt of, or other similar proceedings affecting, the Issuer
     or any of the assets of the Issuer;

          (e)  any invalidity of, or defect or deficiency in, the Capital
     Securities;

          (f)  the settlement or compromise of any obligation guaranteed hereby
     or hereby incurred; or

                                      10
<PAGE>

          (g)  any other circumstance whatsoever that might otherwise constitute
     a legal or equitable discharge or defense of a guarantor, it being the
     intent of this Section 4.3 that the obligations of the Guarantor hereunder
     shall be absolute and unconditional under any and all circumstances.

          There shall be no obligation of the Holders to give notice to, or
obtain consent of, the Guarantor with respect to the happening of any of the
foregoing.

SECTION 4.4  Rights of Holders
             -----------------

          (a)  The Holders of a Majority in liquidation amount of the Capital
     Securities have the right to direct the time, method and place of
     conducting any proceeding for any remedy available to the Guarantee Trustee
     in respect of this Guarantee or to direct the exercise of any trust or
     power conferred upon the Guarantee Trustee under this Guarantee; provided,
                                                                      --------
     however, that (subject to Sections 2.1 and 2.2) the Guarantee Trustee shall
     -------
     have the right to decline to follow any such direction if the Guarantee
     Trustee shall determine that the actions so directed would be unjustly
     prejudicial to the Holders not taking part in such direction or if the
     Guarantee Trustee being advised by counsel determines that the action or
     proceeding so directed may not lawfully be taken or if the Guarantor
     Trustee in good faith by its board of directors or trustees, executive
     committees or a trust committee of directors or trustees and/or Responsible
     Officers shall determine that the action or proceedings so directed would
     involve the Guarantee Trustee in personal liability.

          (b)  Any Holder of Capital Securities may institute a legal proceeding
     directly against the Guarantor to enforce the Guarantee Trustee's rights
     under this Guarantee, without first instituting a legal proceeding against
     the Issuer, the Guarantee Trustee or any other Person.  The Guarantor
     waives any right or remedy to require that any such action be brought first
     against the Issuer, the Guarantee Trustee or any other Person before so
     proceeding directly against the Guarantor.

SECTION 4.5  Guarantee of Payment
             --------------------

          This Guarantee creates a guarantee of payment and not of collection.

SECTION 4.6  Subrogation
             -----------

          The Guarantor shall be subrogated to all (if any) rights of the
Holders of Capital Securities against the Issuer in respect of any amounts paid
to such Holders by the Guarantor under this Guarantee; provided, however, that
                                                       --------  -------
the Guarantor shall not (except to the extent required by applicable provisions
of law) be entitled to enforce or exercise any right that it may acquire by way
of subrogation or any indemnity, reimbursement or other agreement, in all cases
as a result of payment under this Guarantee, if, after giving effect to any such
payment, any amounts are due and unpaid under this Guarantee.  If any amount
shall be paid to the Guarantor in violation of the preceding sentence, the
Guarantor agrees to hold such amount in trust for the Holders and to pay over
such amount to the Holders.

                                      11
<PAGE>

SECTION 4.7  Independent Obligations
             -----------------------

          The Guarantor acknowledges that its obligations hereunder are
independent of the obligations of the Issuer with respect to the Capital
Securities and that the Guarantor shall be liable as principal and as debtor
hereunder to make Guarantee Payments pursuant to the terms of this Guarantee
notwithstanding the occurrence of any event referred to in subsections (a)
through (g), inclusive, of Section 4.3 hereof.

SECTION 4.8  Enforcement.
             -----------

          A Beneficiary may enforce the obligations of the Guarantor contained
in Section 4.1(b) directly against the Guarantor, and the Guarantor waives any
right or remedy to require that any action be brought against the Issuer or any
other person or entity before proceeding against the Guarantor.

          The Guarantor shall be subrogated to all rights (if any) of any
Beneficiary against the Issuer in respect of any amounts paid to the
Beneficiaries by the Guarantor under this Guarantee; provided, however, that the
                                                     --------  -------
Guarantor shall not (except to the extent required by applicable provisions of
law) be entitled to enforce or exercise any rights that it may acquire by way of
subrogation or any indemnity, reimbursement or other agreement, in all cases as
a result of payment under this Guarantee, if, after giving effect to such
payment, any amounts are due and unpaid under this Guarantee.


                                   ARTICLE V
                   LIMITATION OF TRANSACTIONS; SUBORDINATION

SECTION 5.1  Limitation of Transactions
             --------------------------

          So long as any Capital Securities remain outstanding, if (a) there
shall have occurred and be continuing an Event of Default or (b) the Guarantor
shall have selected an Extension Period as provided in the Declaration and such
period, or any extension thereof, shall have commenced and be continuing, then
the Guarantor may not (x) declare or pay any dividends or distributions on, or
redeem, purchase, acquire, or make a liquidation payment with respect to, any of
the Guarantor's capital stock or (y) make any payment of principal of or
interest or premium, if any, on or repay, repurchase or redeem any debt
securities of the Guarantor that rank pari passu in all respects with or junior
in interest to the Debentures (other than (i) payments under this Guarantee,
(ii) repurchases, redemptions or other acquisitions of shares of capital stock
of the Guarantor in connection with any employment contract, benefit plan or
other similar arrangement with or for the benefit of one or more employees,
officers, directors, or consultants, in connection with a dividend reinvestment
or stockholder stock purchase plan or in connection with the issuance of capital
stock of the Guarantor (or securities convertible into or exercisable for such
capital stock) as consideration in an acquisition transaction entered into prior
to occurrence of the Event of Default or the applicable Extension Period, (iii)
as a result of any exchange of conversion of any class or series of the
Guarantor's capital stock (or any capital stock of a subsidiary of the
Guarantor) for any class or series of the Guarantor's capital stock or of any
class or series of the Guarantor's indebtedness for any class

                                      12
<PAGE>

or series of the Guarantor's capital stock, (iv) the purchase of fractional
interests in shares of the Guarantor's capital stock pursuant to the conversion
or exchange provisions of such capital stock or the security being converted or
exchanged, (v) any declaration of a dividend in connection with any
stockholder's rights plan, or the issuance of rights, stock or other property
under any stockholder's rights plan, or the redemption or repurchase of rights
pursuant thereto, or (vi) any dividend in the form of stock, warrants, options
or other rights where the dividend stock or the stock issuable upon exercise of
such warrants, options or other rights is the same stock as that on which the
dividend is being paid or ranks pari passu with or junior to such stock).
                                ---- -----

SECTION 5.2  Ranking
             -------

          This Guarantee will constitute an unsecured obligation of the
Guarantor and will rank subordinate and junior in right of payment to all
present and future Senior Indebtedness (as defined in the Indenture) of the
Guarantor.  By their acceptance thereof, each Holder of Capital Securities
agrees to the foregoing provisions of this Guarantee and the other terms set
forth herein.

          The right of the Guarantor to participate in any distribution of
assets of any of its subsidiaries upon any such subsidiary's liquidation or
reorganization or otherwise is subject to the prior claims of creditors of that
subsidiary, except to the extent the Guarantor may itself be recognized as a
creditor of that subsidiary.  Accordingly, the Guarantor's obligations under
this Guarantee will be effectively subordinated to all existing and future
liabilities of the Guarantor's subsidiaries, and claimants should look only to
the assets of the Guarantor for payments thereunder.  This Guarantee does not
limit the incurrence or issuance of other secured or unsecured debt of the
Guarantor, including Senior Indebtedness of the Guarantor, under any indenture
or agreement that the Guarantor may enter into in the future or otherwise.

                                   ARTICLE VI
                                  TERMINATION

SECTION 6.1  Termination
             -----------

          This Guarantee shall terminate as to the Capital Securities (i) upon
full payment of the Redemption Price or the Special Redemption Price, as the
case may be, of all Capital Securities then outstanding, (ii) upon the
distribution of all of the Debentures to the Holders of all of the Capital
Securities or (iii) upon full payment of the amounts payable in accordance with
the Declaration upon dissolution of the Issuer.  This Guarantee will continue to
be effective or will be reinstated, as the case may be, if at any time any
Holder of Capital Securities must restore payment of any sums paid under the
Capital Securities or under this Guarantee.

                                  ARTICLE VII
                                INDEMNIFICATION

SECTION 7.1  Exculpation
             -----------

          (a)  No Indemnified Person shall be liable, responsible or accountable
     in damages or otherwise to the Guarantor or any Covered Person for any
     loss, damage or claim incurred by reason of any act or omission performed
     or omitted by such Indemnified

                                      13
<PAGE>

     Person in good faith in accordance with this Guarantee and in a manner that
     such Indemnified Person reasonably believed to be within the scope of the
     authority conferred on such Indemnified Person by this Guarantee or by law,
     except that an Indemnified Person shall be liable for any such loss, damage
     or claim incurred by reason of such Indemnified Person's negligence or
     willful misconduct with respect to such acts or omissions.

          (b)  An Indemnified Person shall be fully protected in relying in good
     faith upon the records of the Issuer or the Guarantor and upon such
     information, opinions, reports or statements presented to the Issuer or the
     Guarantor by any Person as to matters the Indemnified Person reasonably
     believes are within such other Person's professional or expert competence
     and who, if selected by such Indemnified Person, has been selected with
     reasonable care by such Indemnified Person, including information,
     opinions, reports or statements as to the value and amount of the assets,
     liabilities, profits, losses, or any other facts pertinent to the existence
     and amount of assets from which Distributions to Holders of Capital
     Securities might properly be paid.

SECTION 7.2  Indemnification
             ---------------

          (a) The Guarantor agrees to indemnify each Indemnified Person for, and
to hold each Indemnified Person harmless against, any and all loss, liability,
damage, claim or expense incurred without negligence or willful misconduct on
the part of the Indemnified Person, arising out of or in connection with the
acceptance or administration of the trust or trusts hereunder, including but not
limited to the costs and expenses (including reasonable legal fees and expenses)
of the Indemnified Person defending itself against, or investigating, any claim
or liability in connection with the exercise or performance of any of the
Indemnified Person's powers or duties hereunder.  The obligation to indemnify as
set forth in this Section 7.2 shall survive the resignation or removal of the
Guarantee Trustee and the termination of this Guarantee.

          (b) Promptly after receipt by an Indemnified Person under this Section
7.2 of notice of the commencement of any action, such Indemnified Person will,
if a claim in respect thereof is to be made against the Guarantor under this
Section 7.2, notify the Guarantor in writing of the commencement thereof; but
the failure so to notify the Guarantor (i) will not relieve the Guarantor from
liability under paragraph (a) above unless and to the extent that the Guarantor
did not otherwise learn of such action and such failure results in the
forfeiture by the Guarantor of substantial rights and defenses and (ii) will
not, in any event, relieve the Guarantor from any obligations to any Indemnified
Person other than the indemnification obligation provided in paragraph (a)
above.  The Guarantor shall be entitled to appoint counsel of the Guarantor's
choice at the Guarantor's expense to represent the Indemnified Person in any
action for which indemnification is sought (in which case the Guarantor shall
not thereafter be responsible for the fees and expenses of any separate counsel
retained by the Indemnified Person or Persons except as set forth below);
provided, however, that such counsel shall be satisfactory to the Indemnified
--------  -------
Person.  Notwithstanding the Guarantor's election to appoint counsel to
represent the Guarantor in an action, the Indemnified Person shall have the
right to employ separate counsel (including local counsel), and the Guarantor
shall bear the reasonable fees, costs and expenses of such separate counsel if
(i) the use of counsel chosen by the Guarantor to

                                      14
<PAGE>

represent the Indemnified Person would present such counsel with a conflict of
interest, (ii) the actual or potential defendants in, or targets of, any such
action include both the Indemnified Person and the Guarantor and the Indemnified
Person shall have reasonably concluded that there may be legal defenses
available to it and/or other Indemnified Person which are different from or
additional to those available to the Guarantor, (iii) the Guarantor shall not
have employed counsel satisfactory to the Indemnified Person to represent the
Indemnified Person within a reasonable time after notice of the institution of
such action or (iv) the Guarantor shall authorize the Indemnified Person to
employ separate counsel at the expense of the Guarantor. The Guarantor will not,
without the prior written consent of the Indemnified Persons, settle or
compromise or consent to the entry of any judgment with respect to any pending
or threatened claim, action, suit or proceeding in respect of which
indemnification or contribution may be sought hereunder (whether or not the
Indemnified Persons are actual or potential parties to such claim or action)
unless such settlement, compromise or consent includes an unconditional release
of each Indemnified Person from all liability arising out of such claim, action,
suit or proceeding.

SECTION 7.3  Compensation; Reimbursement of Expenses
             ---------------------------------------

          The Guarantor agrees:

          (a)  to pay to the Guarantee Trustee from time to time such
compensation for all services rendered by it hereunder as the parties shall
agree to from time to time (which compensation shall not be limited by any
provision of law in regard to the compensation of a trustee of an express
trust); and

          (b)  except as otherwise expressly provided herein, to reimburse the
Guarantee Trustee upon request for all reasonable expenses, disbursements and
advances incurred or made by it in accordance with any provision of this
Guarantee (including the reasonable compensation and the expenses and
disbursements of its agents and counsel), except any such expense, disbursement
or advance as may be attributable to its negligence or willful misconduct.

          The provisions of this Section 7.3 shall survive the resignation or
removal of the Guarantee Trustee and the termination of this Guarantee.

                                  ARTICLE VIII
                                 MISCELLANEOUS

SECTION 8.1  Successors and Assigns
             ----------------------

          All guarantees and agreements contained in this Guarantee shall bind
the successors, assigns, receivers, trustees and representatives of the
Guarantor and shall inure to the benefit of the Holders of the Capital
Securities then outstanding.  Except in connection with any merger or
consolidation of the Guarantor with or into another entity or any sale, transfer
or lease of the Guarantor's assets to another entity, in each case, to the
extent permitted under the Indenture, the Guarantor may not assign its rights or
delegate its obligations under this Guarantee without the prior approval of the
Holders of at least a Majority in liquidation amount of the Capital Securities.

                                      15
<PAGE>

SECTION 8.2  Amendments
             ----------

          Except with respect to any changes that do not adversely affect the
rights of Holders of the Capital Securities in any material respect (in which
case no consent of Holders will be required), this Guarantee may be amended only
with the prior approval of the Holders of not less than a Majority in
liquidation amount of the Capital Securities.  The provisions of the Declaration
with respect to amendments thereof shall apply equally with respect to
amendments of the Guarantee.

SECTION 8.3  Notices
             -------

          All notices provided for in this Guarantee shall be in writing, duly
signed by the party giving such notice, and shall be delivered, telecopied or
mailed by first class mail, as follows:

          (a)  If given to the Guarantee Trustee, at the Guarantee Trustee's
     mailing address set forth below (or such other address as the Guarantee
     Trustee may give notice of to the Holders of the Capital Securities):

               The Bank of New York
               101 Barclay Street, Floor 21W
               New York, NY 10286
               Attention: Corporate Trust Administration
               Telecopy: 212-815-5915

          (b)  If given to the Guarantor, at the Guarantor's mailing address set
     forth below (or such other address as the Guarantor may give notice of to
     the Holders of the Capital Securities and to the Guarantee Trustee):

               Greater Bay Bancorp
               400 Emerson Street, 3/rd/ Floor
               Palo Alto, California  94301
               Attention:  Linda M. Iannone
               Telecopy:  (650) 473-9419

          (c)  If given to any Holder of the Capital Securities, at the address
     set forth on the books and records of the Issuer.

          All such notices shall be deemed to have been given when received in
person, telecopied with receipt confirmed, or mailed by first class mail,
postage prepaid except that if a notice or other document is refused delivery or
cannot be delivered because of a changed address of which no notice was given,
such notice or other document shall be deemed to have been delivered on the date
of such refusal or inability to deliver.

SECTION 8.4  Benefit
             -------

                                      16
<PAGE>

          This Guarantee is solely for the benefit of the Holders of the Capital
Securities and, subject to Section 2.1(a), is not separately transferable from
the Capital Securities.

SECTION 8.5  Governing Law
             -------------

          THIS GUARANTEE SHALL BE GOVERNED BY, AND CONSTRUED IN ACCORDANCE WITH,
THE LAW OF THE STATE OF NEW YORK, WITHOUT REGARD TO CONFLICT OF LAWS PRINCIPLES
THEREOF.

SECTION 8.6  Counterparts
             ------------

          This Guarantee may contain more than one counterpart of the signature
page and this Guarantee may be executed by the affixing of the signature of the
Guarantor and the Guarantee Trustee to any of such counterpart signature pages.
All of such counterpart signature pages shall be read as though one, and they
shall have the same force and effect as though all of the signers had signed a
single signature page.

                                      17
<PAGE>

          THIS GUARANTEE is executed as of the day and year first above written.

                                         GREATER BAY BANCORP,
                                           as Guarantor

                                             /s/ Shawn E. Saunders
                                         By: ___________________________________
                                             Name:  Shawn E. Saunders
                                             Title: Senior Vice President,
                                                    Finance and Accounting

                                         THE BANK OF NEW YORK,
                                           as Guarantee Trustee

                                             /s/ Mary LaGumina
                                         By: ___________________________________
                                             Name:  Mary LaGumina
                                             Title: Vice President