<PAGE>   1
 
                                                                     EXHIBIT 4.4
================================================================================
 
                         TENNESSEE GAS PIPELINE COMPANY
 
                                      AND
 
                           THE CHASE MANHATTAN BANK,
 
                                                                      AS TRUSTEE
 
                               ------------------
 
                          THIRD SUPPLEMENTAL INDENTURE
 
                           DATED AS OF MARCH 13, 1997
 
                                       TO
 
                                   INDENTURE
 
                           DATED AS OF MARCH 4, 1997
 
                               ------------------
 
                         PROVIDING FOR THE ISSUANCE OF
                           7 5/8% DEBENTURES DUE 2037
 
================================================================================
<PAGE>   2
 
  THIRD SUPPLEMENTAL INDENTURE dated as of March 13, 1997, between TENNESSEE GAS
PIPELINE COMPANY, a corporation duly organized and existing under the laws of
the State of Delaware (hereinafter called the "Company"), and THE CHASE
MANHATTAN BANK, a New York banking corporation, as trustee (hereinafter called
the "Trustee").
 
  WHEREAS, the Company has heretofore executed and delivered to the Trustee an
indenture dated as of March 4, 1997 (hereinafter called the "Original
Indenture"), to provide for the issue of an unlimited amount of debentures,
notes and/or other debt obligations of the Company (hereinafter referred to as
the "Securities"), the terms of which are to be determined as set forth in
Section 301 of the Original Indenture; and
 
  WHEREAS, Section 901 of the Original Indenture provides, among other things,
that the Company and the Trustee may enter into indentures supplemental to the
Original Indenture for, among other things, the purpose of setting forth the
terms of Securities of any series; and
 
  WHEREAS, the Company desires to create a series of the Securities in an
aggregate principal amount of $300,000,000 to be designated the "7 5/8%
Debentures Due 2037" (the "Debentures"), and all action on the part of the
Company necessary to authorize the issuance of the Debentures under the Original
Indenture and this Third Supplemental Indenture has been duly taken; and
 
  WHEREAS, all acts and things necessary to make the Debentures, when executed
by the Company and authenticated and delivered by the Trustee as in the Original
Indenture provided, the valid and binding obligations of the Company, and to
constitute these presents a valid and binding supplemental indenture and
agreement according to its terms, have been done and performed;
 
  NOW, THEREFORE, in consideration of the premises and of the mutual covenants
herein contained, and of the acceptance of this trust by the Trustee, and of the
sum of one dollar to the Company duly paid by the Trustee at the execution and
delivery of these presents, and of other valuable consideration the receipt and
sufficiency whereof are hereby acknowledged and in order to authorize the
authentication and delivery of and to set forth the terms of the Debentures,
<PAGE>   3
 
                                        2
 
  IT IS HEREBY COVENANTED, DECLARED AND AGREED by and between the parties
hereto, for the benefit of holders of the Debentures issued under the Indenture,
as follows:
 
                                   ARTICLE 1.
 
                TERMS AND ISSUANCE OF 7 5/8% DEBENTURES DUE 2037
 
  SECTION 1.01 Issue of Debentures. A series of Securities which shall be
designated the "7 5/8% Debentures Due 2037" shall be executed, authenticated and
delivered in accordance with the provisions of, and shall in all respects be
subject to, the terms, conditions and covenants of the Original Indenture,
including without limitation the terms set forth in this Third Supplemental
Indenture (including the form of Debentures referred to in SECTION 1.02 hereof).
The aggregate principal amount of Debentures which may be authenticated and
delivered under the Indenture shall not, except as permitted by the provisions
of the Original Indenture, exceed $300,000,000. The entire amount of Debentures
may forthwith be executed by the Company and delivered to the Trustee and shall
be authenticated by the Trustee and delivered to or upon the order of the
Company pursuant to Section 303 of the Indenture.
 
  SECTION 1.02. Forms of Debentures and Authentication Certificate. The
Debentures initially shall be issuable in the form of one or more Global
Securities. The forms of the Debentures and the Trustee's certificate of
authentication shall be substantially as set forth on Exhibit A hereto.
 
                                   ARTICLE 2.
 
                                 MISCELLANEOUS
 
  SECTION 2.01. Execution as Supplemental Indenture. This Third Supplemental
Indenture is executed and shall be construed as an indenture supplemental to the
Original Indenture and, as provided in the Original Indenture, this Third
Supplemental Indenture forms a part thereof. Except as herein expressly
otherwise defined, the use of the terms and expressions herein is in accordance
with the definitions, uses and constructions contained in the Original
Indenture.
 
  SECTION 2.02. Responsibility for Recitals, Etc. The recitals herein and in the
Debentures (except in the Trustee's certificate of authentication) shall be
taken as the statements of the Company, and the Trustee assumes no
<PAGE>   4
 
                                        3
 
responsibility for the correctness thereof. The Trustee makes no representations
as to the validity or sufficiency of this Third Supplemental Indenture or of the
Debentures. The Trustee shall not be accountable for the use or application by
the Company of the Debentures or of the proceeds thereof.
 
  SECTION 2.03. Provisions Binding on Company's Successors. All the covenants,
stipulations, promises and agreements in this Third Supplemental Indenture
contained by the Company shall bind its successors and assigns whether so
expressed or not.
 
  SECTION 2.04. New York Contract. THIS THIRD SUPPLEMENTAL INDENTURE AND EACH
DEBENTURE SHALL BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH THE LAWS OF THE
STATE OF NEW YORK.
 
  SECTION 2.05. Execution and Counterparts. This Third Supplemental Indenture
may be executed with counterpart signature pages or in any number of
counterparts, each of which shall be an original but such counterparts shall
together constitute but one and the same instrument.
 
  SECTION 2.06. Capitalized Terms. Capitalized terms not otherwise defined in
this Third Supplemental Indenture shall have the respective meanings assigned to
them in the Original Indenture.
<PAGE>   5
 
                                        4
 
  IN WITNESS WHEREOF, said TENNESSEE GAS PIPELINE COMPANY has caused this Third
Supplemental Indenture to be executed in its corporate name by its Chairman of
the Board or its President or one of its Vice Presidents, and said THE CHASE
MANHATTAN BANK has caused this Third Supplemental Indenture to be executed in
its corporate name by one of its Vice Presidents as of March 13, 1997.
 
                                           TENNESSEE GAS PIPELINE COMPANY
 
                                           By      /s/ H. BRENT AUSTIN
                                             -----------------------------------
                                                       H. Brent Austin
                                                    Senior Vice President
 
                                           THE CHASE MANHATTAN BANK
 
                                           By    /s/ RONALD J. HALLERAN
                                             -----------------------------------
                                                     Ronald J. Halleran
                                                    Second Vice President
<PAGE>   6
 
                                                                       EXHIBIT A
 
  THIS SECURITY IS A GLOBAL SECURITY WITHIN THE MEANING OF THE INDENTURE
HEREINAFTER REFERRED TO AND IS REGISTERED IN THE NAME OF A DEPOSITARY OR A
NOMINEE THEREOF. THIS SECURITY MAY NOT BE TRANSFERRED TO, OR REGISTERED OR
EXCHANGED FOR SECURITIES REGISTERED IN THE NAME OF, ANY PERSON OTHER THAN THE
DEPOSITARY OR A NOMINEE THEREOF AND NO SUCH TRANSFER MAY BE REGISTERED, EXCEPT
IN THE LIMITED CIRCUMSTANCES DESCRIBED IN THE INDENTURE. EVERY SECURITY
AUTHENTICATED AND DELIVERED UPON REGISTRATION OF TRANSFER OF, OR IN EXCHANGE FOR
OR IN LIEU OF, THIS SECURITY SHALL BE A GLOBAL SECURITY SUBJECT TO THE
FOREGOING, EXCEPT IN SUCH LIMITED CIRCUMSTANCES.
 
  UNLESS THIS SECURITY IS PRESENTED BY AN AUTHORIZED REPRESENTATIVE OF THE
DEPOSITORY TRUST COMPANY, A NEW YORK CORPORATION, TO THE COMPANY OR ITS AGENT
FOR REGISTRATION OF TRANSFER, EXCHANGE OR PAYMENT, AND ANY SECURITY ISSUED IS
REGISTERED IN THE NAME OF CEDE & CO. OR SUCH OTHER NAME AS IS REQUESTED BY AN
AUTHORIZED REPRESENTATIVE OF THE DEPOSITORY TRUST COMPANY (AND ANY PAYMENT IS
MADE TO CEDE & CO. OR TO SUCH OTHER ENTITY AS IS REQUESTED BY AN AUTHORIZED
REPRESENTATIVE OF THE DEPOSITORY TRUST COMPANY), ANY TRANSFER, PLEDGE OR OTHER
USE HEREOF FOR VALUE OR OTHERWISE BY OR TO ANY PERSON IS WRONGFUL SINCE THE
REGISTERED OWNER HEREOF, CEDE & CO., HAS AN INTEREST HEREIN.
<PAGE>   7
 
                                        2
 
                         TENNESSEE GAS PIPELINE COMPANY
 
                           7 5/8% DEBENTURES DUE 2037
 
NO.                                                               U.S.$
------------------------                                          -------------
CUSIP NO. 880451 AU3
 
  TENNESSEE GAS PIPELINE COMPANY, a corporation duly incorporated and existing
under the laws of Delaware (herein called the "Company", which term includes any
successor Person under the Indenture hereinafter referred to), for value

received, hereby promises to pay to ------------------, or registered assigns,
the principal sum of ------------------------------------ United States Dollars
on April 1, 2037, and to pay interest thereon from March 13, 1997, or from the
most recent Interest Payment Date to which interest has been paid or duly
provided for, semi-annually on April 1 and October 1 in each year, commencing
October 1, 1997, at the rate of 7 5/8% per annum, until the principal hereof is
paid or made available for payment. The interest so payable, and punctually paid
or duly provided for, on any Interest Payment Date will, as provided in such
Indenture, be paid to the Person in whose name this Security (or one or more
Predecessor Securities) is registered at the close of business on the Regular
Record Date for such interest, which shall be the March 15 or September 15
(whether or not a Business Day), as the case may be, next preceding such
Interest Payment Date. Any such interest not so punctually paid or duly provided
for will forthwith cease to be payable to the Holder on such Regular Record Date
and may either be paid to the Person in whose name this Security (or one or more
Predecessor Securities) is registered at the close of business on a Special
Record Date for the payment of such Defaulted Interest to be fixed by the
Trustee, notice of which shall be given to Holders of Securities of this series
not less than 10 days prior to such Special Record Date, or be paid at any time
in any other lawful manner not inconsistent with the requirements of any
securities exchange on which the Securities of this series may be listed, and
upon such notice as may be required by such exchange, all as more fully provided
in such Indenture.
 
  Payment of the principal of and interest on this Security will be made by
transfer of immediately available funds to a bank account in New York, New York,
designated by the Holder in such coin or currency of the United States of
America as at the time of payment is legal tender for payment of public and
private debts.
<PAGE>   8
 
                                        3
 
  Reference is hereby made to the further provisions of this Security set forth
on the reverse hereof, which further provisions shall for all purposes have the
same effect as if set forth at this place.
 
  Unless the certificate of authentication hereon has been executed by the
Trustee referred to on the reverse hereof by manual signature, this Security
shall not be entitled to any benefit under the Indenture or be valid or
obligatory for any purpose.
 
  IN WITNESS WHEREOF, the Company has caused this instrument to be duly
executed.
 
Dated:
 
                                           TENNESSEE GAS PIPELINE COMPANY
 
                                           By:
                                              ----------------------------------
                                              H. Brent Austin
                                              Senior Vice President
 
  This is one of the Securities of the series designated therein referred to in
the within-mentioned Indenture.
 
                                           THE CHASE MANHATTAN BANK,
                                             AS TRUSTEE
 
                                           By:
                                              ----------------------------------
                                              Authorized Officer
<PAGE>   9
 
                                        4
 
                         TENNESSEE GAS PIPELINE COMPANY
 
                           7 5/8% DEBENTURES DUE 2037
 
  This Security is one of a duly authorized issue of securities of the Company
(the "Securities"), issued and to be issued in one or more series under an
Indenture dated as of March 4, 1997 (the "Indenture"), between the Company and
The Chase Manhattan Bank, as Trustee (the "Trustee", which term includes any
successor trustee under the Indenture), to which Indenture and all indentures
supplemental thereto reference is hereby made for a statement of the respective
rights, limitations of rights, obligations, duties and immunities thereunder of
the Company, the Trustee and the Holders of the Securities and of the terms upon
which the Securities are, and are to be, authenticated and delivered. As
provided in the Indenture, the Securities may be issued in one or more series,
which different series may be issued in various aggregate principal amounts, may
mature at different times, may bear interest, if any, at different rates, may be
subject to different redemption provisions, if any, may be subject to different
sinking, purchase or analogous funds, if any, may be subject to different
covenants and Events of Default and may otherwise vary as in the Indenture
provided or permitted. This Security is one of the series designated on the face
hereof, limited in aggregate principal amount to U.S.$300,000,000.
 
  The Securities of this series are not redeemable prior to Stated Maturity.
 
  If an Event of Default with respect to Securities of this series shall occur
and be continuing, the principal of the Securities of this series may be
declared due and payable in the manner and with the effect provided in the
Indenture.
 
  The Indenture permits, with certain exceptions as therein provided, the
amendment thereof and the modification of the rights and obligations of the
Company and the rights of the Holders of the Securities of each series to be
affected under the Indenture at any time by the Company and the Trustee with the
consent of the Holders of a majority in aggregate principal amount of the
Outstanding Securities of all series to be affected (voting as one class). The
Indenture also contains provisions permitting the Holders of a majority in
aggregate principal amount of the Outstanding Securities of all affected series
(voting as one class), on behalf of the Holders of all Securities of such
series, to waive compliance by the Company with certain provisions of the
Indenture. The Indenture permits, with certain exceptions as therein provided,
the Holders of a majority in principal amount of Securities of all affected
series
<PAGE>   10
 
                                        5
 
then Outstanding (voting as a single class) to waive past defaults under the
Indenture with respect to such Securities and their consequences. Any such
consent or waiver by the Holder of this Security shall be conclusive and binding
upon such Holder and upon all future Holders of this Security and of any
Security issued upon the registration of transfer hereof or in exchange herefor
or in lieu hereof, whether or not notation of such consent or waiver is made
upon this Security.
 
  As provided in and subject to the provisions of the Indenture, the Holder of
this Security shall not have the right to institute any proceeding with respect
to the Indenture or for the appointment of a receiver or trustee or for any
other remedy thereunder, unless such Holder shall have previously given the
Trustee written notice of a continuing Event of Default with respect to the
Securities of this series, the Holders of not less than 25% in principal amount
of the Securities of all affected series at the time Outstanding (treated as a
single class) shall have made written request to the Trustee to institute
proceedings in respect of such Event of Default as Trustee and offered the
Trustee reasonable indemnity and the Trustee shall not have received from the
Holders of a majority in principal amount of Securities of this series at the
time Outstanding a direction inconsistent with such request, and shall have
failed to institute any such proceeding, for 60 days after receipt of such
notice, request and offer of indemnity. The foregoing shall not apply to any
suit instituted by the Holder of this Security for the enforcement of any
payment of principal hereof or interest hereon on or after the respective due
dates expressed herein.
 
  No reference herein to the Indenture and no provision of this Security or of
the Indenture shall alter or impair the obligation of the Company, which is
absolute and unconditional, to pay the principal of and interest on this
Security at the times, place(s) and rate, and in the coin or currency, herein
prescribed.
 
  This Global Security or portion hereof may not be exchanged for Definitive
Securities of this series except in the limited circumstances provided in the
Indenture.
 
  The holders of beneficial interests in this Global Security will not be
entitled to receive physical delivery of Definitive Securities except as
described in the Indenture and will not be considered the Holders thereof for
any purpose under the Indenture.
 
  The Securities of this series are issuable only in registered form without
coupons in denominations of U.S. $1,000 and any integral multiple thereof. As
<PAGE>   11
 
                                        6
 
provided in the Indenture and subject to certain limitations therein set forth,
Securities of this series are exchangeable for a like aggregate principal amount
of Securities of this series and of like tenor of a different authorized
denomination, as requested by the Holder surrendering the same.
 
  No service charge shall be made for any such registration of transfer or
exchange, but the Company may require payment of a sum sufficient to cover any
tax or other governmental charge payable in connection therewith.
 
  Prior to due presentment of this Security for registration of transfer, the
Company, the Trustee and any agent of the Company or the Trustee may treat the
Person in whose name this Security is registered as the owner hereof for all
purposes, whether or not this Security be overdue, and neither the Company, the
Trustee nor any such agent shall be affected by notice to the contrary.
 
  No recourse under or upon any obligation, covenant or agreement of or
contained in the Indenture or of or contained in any Security, or for any claim
based thereon or otherwise in respect thereof, or in any Security, or because of
the creation of any indebtedness represented thereby, shall be had against any
incorporator, stockholder, officer or director, as such, past, present or
future, of the Company or of any successor Person, either directly or through
the Company or any successor Person, whether by virtue of any constitution,
statute or rule of law, or by the enforcement of any assessment, penalty or
otherwise; it being expressly understood that all such liability is hereby
expressly waived and released by the acceptance hereof and as a condition of,
and as part of the consideration for, the Securities and the execution of the
Indenture.
 
  The Indenture provides that the Company (a) will be discharged from any and
all obligations in respect of the Securities (except for certain obligations
described in the Indenture), or (b) need not comply with certain restrictive
covenants of the Indenture, in each case if the Company deposits, in trust, with
the Trustee money or U.S. Government Obligations (or a combination thereof)
which through the payment of interest thereon and principal thereof in
accordance with their terms will provide money, in an amount sufficient to pay
all the principal of and interest on the Securities, but such money need not be
segregated from other funds except to the extent required by law.
 
  THIS SECURITY SHALL BE GOVERNED BY AND CONSTRUED IN ACCORDANCE WITH THE LAWS
OF THE STATE OF NEW YORK.
 
  All terms used in this Security which are defined in the Indenture shall have
the meanings assigned to them in the Indenture.