<PAGE>   1
                                                                     EXHIBIT 4.1

                             STOCK OPTION AGREEMENT

         STOCK OPTION AGREEMENT, dated as of June 21, 2000 (the "STOCK OPTION
AGREEMENT"), between Texas Instruments Incorporated, a Delaware corporation
("PARENT"), and Burr-Brown Corporation, a Delaware corporation (the "COMPANY").

         WHEREAS, Parent, Burma Acquisition Corp., a Delaware corporation and a
wholly owned subsidiary of Parent ("MERGER SUB"), and the Company are parties to
that certain Agreement and Plan of Merger, dated as of the date hereof (the
"MERGER AGREEMENT"), which provides, among other things, that Merger Sub, on the
terms and subject to the conditions thereof, will merge with and into the
Company with the Company surviving as a wholly owned subsidiary of Parent;

         WHEREAS, as a condition to its willingness to enter into the Merger
Agreement, Parent has requested that the Company grant to Parent an option to
purchase up to 11,236,702 shares of common stock, par value $0.01 per share
("COMMON STOCK") of the Company, upon the terms and subject to the conditions
hereof; and

         WHEREAS, in order to induce Parent to enter into the Merger Agreement,
the Company is willing to grant Parent the requested option.

         NOW, THEREFORE, in consideration of the premises and the mutual
covenants and agreements set forth herein, the parties hereto agree as follows:

         1.       The Option; Exercise; Adjustments.

                  (a) Subject to the other terms and conditions set forth
herein, the Company hereby grants to Parent an irrevocable option (the "OPTION")
to purchase up to 11,236,702 shares of Common Stock (the "SHARES"). The purchase
price per Share (the "PURCHASE PRICE") shall be $112.94. The Purchase Price and
the number of Shares shall be subject to adjustment as provided in Section 1(c)
hereof.

                  (b) Parent may exercise the Option with respect to any or all
of the Option Shares at any one time or from time to time, subject to the
provisions of Section 1(c) hereof, upon the occurrence of an Exercise Event (as
defined below). Subject to the last sentence of this Section 1(b), the Option
will terminate and be of no further force and effect upon the earliest to occur
of (i) the Effective Time (as defined in the Merger Agreement), (ii) 90 days
after the first occurrence of an Exercise Event, and (iii) the termination of
the Merger Agreement in accordance with its terms so long as, in the case of
this clause (iii), no Exercise Event has occurred or could still occur under
Section 8.5(b) of the Merger Agreement, in which case the Option will terminate
on the later of (x) 90 days following the time such termination fee becomes
unconditionally payable and (y) the expiration of the period in which an
Exercise Event could occur pursuant to Section 8.5(b) of the Merger Agreement.
"Exercise Event" means any event as a result of which Parent is unconditionally
entitled to receive a termination fee pursuant to Section 8.5(b) of the Merger
Agreement. Notwithstanding the termination of
<PAGE>   2
the Option, Parent shall be entitled to purchase the Shares with respect to
which it has exercised the Option in accordance with the terms hereof prior to
the termination of the Option.

                  (c) (i) In the event Parent is entitled to and wishes to
exercise the Option, Parent shall send a written notice to the Company (the
"STOCK EXERCISE NOTICE") specifying a date (subject to the HSR Act (as defined
below)) not later than 20 business days and not earlier than three business days
following the date such notice is given for the closing of such purchase and
specifying the number of Shares Parent wishes to purchase.

                      (ii) In the event of any change in the number of issued
and outstanding shares of Common Stock by reason of any stock dividend, stock
split, split-up, recapitalization, merger (other than the Merger) or other
change in the corporate or capital structure of the Company, the number of
Shares subject to this Option and the purchase price per Share shall be
appropriately adjusted to restore the Parent to its rights hereunder, including
its right to purchase Shares representing approximately 19.9% of the common
stock of the Company that is issued and outstanding on the date hereof (after
giving effect to the foregoing adjustments) at an aggregate purchase price equal
to the Purchase Price multiplied by 11,236,702.

                      (iii) In the event that Company shall enter into an
agreement to: (A) consolidate with or merge into any person, other than Parent
or one of its subsidiaries, and Company shall not be the continuing or surviving
corporation of such consolidation or merger; (B) permit any person, other than
Parent or one of its subsidiaries, to merge into Company and Company shall be
the continuing or surviving corporation, but, in connection with such merger,
the then-outstanding shares of Common Stock shall be changed into or exchanged
for stock or other securities of Company or any other person or cash or any
other property, or the outstanding shares of Common Stock immediately prior to
such merger shall after such merger represent less than 50% of the outstanding
shares and share equivalents of the merged company; or (C) sell or otherwise
transfer all or substantially all of its assets to any person, other than Parent
or one of its subsidiaries, then, and in each such case, the agreement governing
such transaction shall make proper provisions so that upon the consummation of
any such transaction, and upon the terms and conditions set forth herein, Parent
shall receive for each Share of Common Stock with respect to which the Option
has not been exercised in an amount of consideration in the form of and equal to
the per share amount of consideration that would be received by the holder of
one share of Common Stock less the Purchase Price to the extent the Option is
then exercisable in accordance with the terms and conditions hereof (and, in the
event of an election or similar arrangement with respect to the type of
consideration to be received by the holders of Common Stock, subject to the
foregoing, proper provision shall be made so that the holder of the Option would
have the same election or similar rights as would the holder of the number of
shares of Common Stock for which the Option is then exercisable).

                  (d) At any time the Option is exercisable pursuant to the
terms of Section 1(b) hereof, Parent may elect, in lieu of exercising the Option
to purchase


                                       2
<PAGE>   3
Shares provided in Section 1(a) hereof, to send a written notice to
the Company (the "CASH EXERCISE NOTICE") specifying a date not later than 20
business days and not earlier than 10 business days following the date such
notice is given on which date the Company shall pay to Parent an amount in cash
equal to the Spread (as hereinafter defined) multiplied by all or such portion
of the Shares subject to the Option as Parent shall specify in such Cash
Exercise Notice. As used herein "SPREAD" shall mean the excess, if any, over the
Purchase Price of the higher of (i) if applicable, the highest price per share
of Common Stock paid or proposed to be paid by any person pursuant to a
definitive agreement executed by the Company with respect to an Acquisition
Proposal (the "ALTERNATIVE PURCHASE PRICE") or (ii) the average closing price,
for the five trading days ending on the trading day immediately preceding the
date of the Cash Exercise Notice, per share of Common Stock as reported on the
Nasdaq National Market (the "CLOSING PRICE"). If the Alternative Purchase Price
includes any property other than cash, the Alternative Purchase Price shall be
the sum of (i) the fixed cash amount, if any, included in the Alternative
Purchase Price plus (ii) the fair market value of such other property. If such
other property consists of securities with an existing public trading market,
the average of the closing prices (or the average of the closing bid and asked
prices if closing prices are unavailable) for such securities in their principal
public trading market on the five trading days ending on the trading day
immediately preceding the date of the Cash Exercise Notice shall be deemed to
equal the fair market value of such property. If such other property consists of
something other than cash or securities with an existing public trading market
and, as of the payment date for the Spread, agreement on the value of such other
property has not been reached between the parties hereto, the Alternative
Purchase Price shall be deemed to equal the Closing Price. Upon exercise of its
right to receive cash pursuant to the exercise of the Option, the obligations of
the Company to deliver Shares pursuant to Section 3 shall be terminated with
respect to such number of Shares for which Parent shall have elected to be paid
the Spread.

         2. Conditions to Delivery of Shares. The Company's obligation to
deliver Shares upon exercise of the Option is subject only to the fulfillment of
the following conditions:

            (i) No preliminary or permanent injunction or other order issued by
any federal or state court of competent jurisdiction in the United States
prohibiting the delivery of the Shares shall be in effect; and

            (ii) Any applicable waiting periods under the Hart-Scott-Rodino
Antitrust Improvements Act of 1976 (the "HSR ACT") shall have expired or been
terminated; and

            (iii) All consents, approvals, orders, notifications, filings or
authorizations, the failure of which to make or obtain would have the effect of
making the issuance of Shares to Parent illegal ("OTHER REQUISITE CONSENTS"),
shall have been made or obtained.


                                       3
<PAGE>   4
         3. The Closing.

            (a) Any closing hereunder shall take place on the date specified by
Parent in its Stock Exercise Notice or Cash Exercise Notice, as the case may be,
at such reasonable time and place as may be indicated in the Stock Exercise
Notice or the Cash Exercise Notice, as applicable, or at the election of the
Company at 10:00 A.M., local time, at the offices of Weil, Gotshal & Manges LLP,
100 Crescent Court, Suite 1300, Dallas, Texas, or, if the conditions set forth
in Section 2(i), 2(ii) or 2(iii) have not then been satisfied, on the second
business day following the satisfaction of such conditions, or at such other
time and place as the parties hereto may agree (the "CLOSING DATE"). On the
Closing Date, (i) in the event of a closing pursuant to Section 1(c) hereof, the
Company will deliver to Parent a certificate or certificates, duly endorsed (or
accompanied by duly executed stock powers), representing the Shares in the
denominations designated by Parent in its Stock Exercise Notice and Parent will
purchase such Shares from the Company at a cash price per Share equal to the
Purchase Price or (ii) in the event of a closing pursuant to Section 1(d)
hereof, the Company will deliver to Parent cash in an amount determined pursuant
to Section 1(d) hereof. Any payment of cash made by Parent to the Company, or by
the Company to Parent, pursuant to this Stock Option Agreement shall be made by
wire transfer of immediately available funds to a bank designated by the party
receiving such funds.

            (b) The certificates representing the Shares may bear an appropriate
legend relating to the fact that such Shares have not been registered under the
Securities Act of 1933, as amended (the "SECURITIES ACT").

            4. Representations and Warranties of the Company. The Company
represents and warrants to Parent that, except as contemplated by the Merger
Agreement, (a) the Company is a corporation duly organized, validly existing and
in good standing under the laws of the State of Delaware and has the requisite
corporate power and authority to enter into and perform this Stock Option
Agreement; (b) the execution and delivery of this Stock Option Agreement by the
Company and the consummation by it of the transactions contemplated hereby have
been duly authorized by the Board of Directors of the Company and this Stock
Option Agreement has been duly and validly executed and delivered by a duly
authorized officer of the Company and will constitute a valid and binding
obligation of the Company subject to bankruptcy, insolvency, fraudulent
transfer, reorganization, moratorium and similar laws of general applicability
relating to or affecting creditors' rights and to general equity principles; (c)
the Company has taken all necessary corporate action to authorize and reserve
the Shares issuable upon exercise of the Option and the Shares, when issued and
delivered by the Company upon exercise of the Option, will be duly authorized,
validly issued, fully paid and non-assessable and free of preemptive rights; (d)
except as otherwise required by the HSR Act or for the Other Requisite Consents,
the execution and delivery of this Stock Option Agreement by the Company and the
consummation by it of the transactions contemplated hereby do not require the
consent, waiver, approval or authorization of or any filing with any person or
public authority and will not violate, result in a breach of or the acceleration
of any obligation under, or constitute a default under, any provision of any
charter or bylaw, indenture, mortgage, lien, lease, agreement, contract,
instrument, order,


                                       4
<PAGE>   5
law, rule, regulation, judgment, ordinance, or decree, or restriction by which
the Company or any of its subsidiaries or any of their respective properties or
assets is bound, except where the failure to obtain such consent, waiver,
approval or authorization or make such filing, or where such breach,
acceleration or default, is not reasonably expected to have a Material Adverse
Effect (as defined in the Merger Agreement) on the Company and its subsidiaries
taken as a whole or on the ability of the Company to consummate the transactions
contemplated hereby; and (e) the board of directors of the Company has taken all
action required so that the restrictions contained in Section 203 of the DGCL
applicable to a "business combination" (as defined in DGCL Section 203) will not
apply to the execution, delivery, or performance of this Agreement or the
consummation of the transactions contemplated by hereby.

                  5. Representations and Warranties of the Parent. Parent
represents and warrants to the Company that (a) Parent is a corporation duly
organized, validly existing and in good standing under the laws of the State of
Delaware and has the requisite corporate power and authority to enter into and
perform this Stock Option Agreement; (b) the execution and delivery of this
Stock Option Agreement by Parent and the consummation by it of the transactions
contemplated hereby have been duly authorized by all necessary corporate action
on the part of Parent and this Stock Option Agreement has been duly and validly
executed and delivered by a duly authorized officer of Parent and will
constitute a valid and binding obligation of Parent enforceable in accordance
with its terms, subject to bankruptcy, insolvency, fraudulent transfer,
reorganization, moratorium and similar laws of general applicability relating to
or affecting creditors' rights and to general equity principles; and (c) Parent
is acquiring the Option and, if and when it exercises the Option, will be
acquiring the Shares issuable upon the exercise thereof for its own account and
not with a view to distribution or resale in any manner which would be in
violation of the Securities Act.

                  6. Listing of Shares; HSR Act Filings; Governmental Consents.
Subject to applicable law and the rules and regulations of the Nasdaq National
Market (or any other national securities exchange or quotation system on which
the Common Stock is then listed) (as applicable, the "STOCK EXCHANGE"), when the
Option becomes exercisable hereunder, the Company will promptly file an
application to list the Shares on the Stock Exchange and will use all reasonable
efforts to effect all necessary filings by the Company under the HSR Act. Each
of the parties hereto will use all reasonable efforts to obtain consents of all
third parties and governmental authorities (including any Other Requisite
Consents), if any, necessary to the consummation of the transactions
contemplated hereby.

                  7. Registration Rights.

                     (a) In the event that Parent shall desire to sell any of
the Shares within two years after the purchase of such Shares pursuant hereto,
and such sale requires, in the reasonable opinion of counsel to Parent,
registration of such Shares under the Securities Act, the Company will cooperate
with Parent and any underwriters in registering such Shares for resale,
including, without limitation, promptly filing a registration statement which
complies with the requirements of applicable federal and


                                       5
<PAGE>   6
state securities laws, entering into an underwriting agreement with such
underwriters upon such terms and conditions as are customarily contained in
underwriting agreements with respect to secondary distributions; provided that
the Company shall not be required to have declared effective more than two
registration statements hereunder and shall be entitled to delay the filing or
effectiveness of any registration statement for up to 120 days if the offering
would, in the good faith judgment of the Board of Directors of the Company,
require premature disclosure of any material corporate development or otherwise
interfere with or adversely affect any pending or proposed offering of
securities of the Company or any other material transaction involving the
Company. Parent shall use its reasonable efforts to cause, and to cause any
underwriters of any sale or disposition to cause, any sale or disposition
pursuant to such registration statement to be effected on a widely disseminated
basis so that upon consummation thereof no purchaser or transferee will own
beneficially more than 2.0% of the then outstanding voting power of the Company.

                     (b) If the Common Stock is registered pursuant to the
provisions of this Section 7, the Company agrees (i) to furnish copies of the
registration statement and the prospectus relating to the Shares covered thereby
in such numbers as Parent may from time to time reasonably request and (ii) if
any event shall occur as a result of which it becomes necessary to amend or
supplement any registration statement or prospectus, to prepare and file under
the applicable securities laws such amendments and supplements as may be
necessary to keep available for at least 90 days a prospectus covering the
Common Stock meeting the requirements of such securities laws, and to furnish
Parent such numbers of copies of the registration statement and prospectus as
amended or supplemented as may reasonably be requested. Parent will provide
information reasonably requested by the Company for inclusion in any
registration statement to be filed pursuant to this Section 7. The Company shall
bear all costs of the registration, including, but not limited to, all
registration and filing fees, printing expenses, and fees and disbursements of
counsel and accountants for the Company, except that Parent shall pay the fees
and disbursements of its counsel, and the underwriting fees and selling
commissions applicable to the Shares sold by Parent. In connection with any
registration pursuant to this Section 7, Parent and the Company shall provide
each other and each underwriter of the offering with customary representations,
warranties and covenants, including indemnification and contribution. If a
requested registration pursuant to this Section 7 involves an underwritten
offering, the underwriter or underwriters thereof shall be a nationally
recognized firm or firms selected by the Company, which firm or firms shall be
reasonably satisfactory to Parent.

         8. Expenses. Each party hereto shall pay its own expenses incurred in
connection with this Stock Option Agreement.

         9. Specific Performance. The Company acknowledges that if the Company
fails to perform any of its obligations under this Stock Option Agreement
immediate and irreparable harm or injury would be caused to Parent for which
money damages would not be an adequate remedy. In such event, the Company agrees
that Parent shall have the right, in addition to any other rights it may have,
to specific performance of this Stock Option Agreement. Accordingly, if Parent
should institute an

                                       6
<PAGE>   7
action or proceeding seeking specific enforcement of the provisions hereof, the
Company hereby waives the claim or defense that Parent has an adequate remedy at
law and hereby agrees not to assert in any such action or proceeding the claim
or defense that such a remedy at law exists. The Company further agrees to waive
any requirements for the securing or posting of any bond in connection with
obtaining any such equitable relief.

         10. Profit Limitation.

             (a) Notwithstanding any other provision of this Stock Option
Agreement, in no event shall Parent's Total Profit (as hereinafter defined)
exceed $255,090,070 and, if it otherwise would exceed such amount, Parent, at
its sole election, shall either (a) deliver to the Company for cancellation
Shares previously purchased by Parent, (b) pay cash to the Company or (c)
undertake any combination thereof, so that Parent's Total Profit shall not
exceed $255,090,070 after taking into account the foregoing actions.

             (b) Notwithstanding any other provision of this Stock Option
Agreement, this Option may not be exercised for a number of Shares as would, as
of the date of the Stock Exercise Notice, result in a Notional Total Profit (as
defined below) of more than $255,090,070 and, if exercise of the Option
otherwise would exceed such amount, Parent, at its discretion, may increase the
Purchase Price for that number of Shares set forth in the Stock Exercise Notice
so that the Notional Total Profit shall not exceed $255,090,070; provided,
however, that nothing in this sentence shall restrict any exercise of the Option
permitted hereby on any subsequent date at the Purchase Price set forth in
Section 1(a) hereof.

             (c) As used herein, the term "TOTAL PROFIT" shall mean the
aggregate amount (before taxes) of the following: (i) the amount of cash
received by Parent pursuant to Section 1(d), (ii) (x) the cash amounts or the
fair market value of any property received by Parent pursuant to the sale of
Shares (or any other securities into which such Shares are converted or
exchanged), less (y) Parent's Purchase Price for such Shares, and (iii) any fees
received pursuant to Section 8.5(b) of the Merger Agreement.

             (d) As used herein, the term "NOTIONAL TOTAL PROFIT" with respect
to any number of Shares as to which Parent may propose to exercise this Option
shall be the Total Profit determined as of the date of the Stock Exercise Notice
assuming that this Option were exercised on such date for such number of Shares
and assuming that such Shares, together with all other Shares held by Parent and
its affiliates as of such date, were sold for cash at the closing market price
for the Common Stock as of the close of business on the preceding trading day
(less customary brokerage commissions).

         11. Transfers. The Shares may not be sold, assigned, transferred, or
otherwise disposed of except (i) in an underwritten public offering as provided
in Section 7 hereof or (ii) to any purchaser or transferee who, to Parent's
knowledge, would immediately following such sale, assignment, transfer or
disposal, beneficially own more than 2.0% of the then outstanding voting power
of the Company; provided, however, that

                                       7
<PAGE>   8
Parent shall be permitted to sell any Shares if such sale is made pursuant to a
tender or exchange offer.

         12. Notice. All notices, requests, demands and other communications
hereunder shall be in writing and shall be deemed to have been fully given if
(i) delivered personally; (ii) sent by certified or registered mail, return
receipt requested; (iii) sent by overnight courier for delivery on the next
business day; or (iv) sent by confirmed facsimile, provided that a hard copy of
all such materials is thereafter sent within 24 hours in the manner described in
clauses (i), (ii) or (iii), to the parties at the following addresses or at such
other addresses as shall be specified by the parties by like notice:

                  If to Parent:

                           Texas Instruments Incorporated
                           7839 Churchill Way, M/S 3995
                           Dallas, Texas  75251

                                     -  or  -

                           P.O. Box 650311, M/S 3995
                           Dallas, Texas  75265
                           Attention:  Charles D. Tobin
                           Facsimile No.:  (972) 917-3804

                  With copies to:

                           Texas Instruments Incorporated
                           12500 TI Boulevard, M/S 8658
                           Dallas, Texas  75243

                                     -  or  -

                           P.O. Box 660199, M/S 8658
                           Dallas, Texas  75266
                           Attention:  Joseph F. Hubach, Esq.
                           Facsimile No.:  (972) 480-5061

                                    and

                           Weil, Gotshal & Manges LLP
                           100 Crescent Court, Suite 1300
                           Dallas, Texas  75201
                           Attention:  R. Scott Cohen
                           Facsimile No.:  (214) 746-7777


                                       8
<PAGE>   9
                  If to the Company:

                           Burr-Brown Corporation
                           6730 South Tucson Boulevard
                           Tucson, Arizona  85706
                           Attention:  Syrus P. Madavi
                           Facsimile No.:  (520) 746-7279

                  With a copy to:

                           Snell & Wilmer, L.L.P.
                           One Arizona Center
                           Phoenix, Arizona 85004-2202
                           Attention:  Steven D. Pidgeon, Esq.
                           Facsimile No.:  (602) 382-6070

         Notices provided in accordance with this Section 12 shall be deemed
delivered (i) on the date of personal delivery, (ii) four business days after
deposit in the mail, (ii) one business day after delivery to an overnight
courier, or (iv) on the date of confirmation of the facsimile transmission, as
the case may be.

             13. Parties in Interest. This Stock Option Agreement shall inure to
the benefit of and be binding upon the parties named herein and their respective
successors and permitted assigns; provided, however, that such successor in
interest or permitted assigns shall agree to be bound by the provisions of this
Stock Option Agreement. Nothing in this Stock Option Agreement, express or
implied, is intended to confer upon any person other than the Company or Parent,
or their successors or assigns, any rights or remedies under or by reason of
this Stock Option Agreement.

             14. Entire Agreement; Amendments. This Stock Option Agreement,
together with the Merger Agreement and the other documents referred to therein,
contains the entire agreement between the parties hereto with respect to the
subject matter hereof and supersedes all prior and contemporaneous agreements
and understandings, oral or written, with respect to such transactions. This
Stock Option Agreement may not be changed, amended or modified orally, but may
be changed only by an agreement in writing signed by the party against whom any
waiver, change, amendment, modification or discharge may be sought.

             15. Assignment. No party to this Stock Option Agreement may assign
any of its rights or obligations under this Stock Option Agreement without the
prior written consent of the other party hereto.

             16. Headings. The section headings herein are for convenience only
and shall not affect the construction of this Stock Option Agreement.

             17. Counterparts. This Stock Option Agreement may be executed in
any number of counterparts, each of which, when executed, shall be deemed to be
an original and all of which together shall constitute one and the same
document.


                                       9
<PAGE>   10
             18. Governing Law. This Stock Option Agreement shall be governed by
and construed in accordance with the laws of the State of Delaware, without
giving effect to the choice of law principles thereof.

             19. Severability. If any term, provision, covenant or restriction
of this Stock Option Agreement is held by a court of competent jurisdiction to
be invalid, void or unenforceable, the remainder of the terms, provisions,
covenants and restrictions of this Stock Option Agreement shall remain in full
force and effect and shall in no way be affected, impaired or invalidated.


            [THIS REMAINDER OF THIS PAGE IS INTENTIONALLY LEFT BLANK]



                                       10
<PAGE>   11
         IN WITNESS WHEREOF, the Parent and the Company have caused this Stock
Option Agreement to be duly executed and delivered on the day and year first
above written.

                            TEXAS INSTRUMENTS INCORPORATED


                            By: /s/ M. Samuel Self
                               --------------------------------------------
                            Name:   M. Samuel Self
                            Title:  Senior Vice President and Controller

                            BURR-BROWN CORPORATION


                            By:  /s/ Syrus P. Madavi
                               --------------------------------------------
                            Name:      Syrus P. Madavi
                            Title:     Treasurer



                                       11