<PAGE>   1
           

                       SECURITIES AND EXCHANGE COMMISSION

                             WASHINGTON, D.C. 20549

                                   FORM 10-Q



          [X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE
                        SECURITIES EXCHANGE ACT OF 1934


                  For the Quarterly Period Ended June 30, 1998

                                       OR


          [ ] TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE
                        SECURITIES EXCHANGE ACT OF 1934


  For the Transition Period from ___________________ to _____________________


                         Commission File Number 0-13546

                               ------------------


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
             ------------------------------------------------------
             (Exact Name of Registrant as Specified in Its Charter)


           Delaware                                              41-1464066
-------------------------------                           ----------------------
(State or Other Jurisdiction of                              (I.R.S. Employer
 Incorporation or Organization)                           Identification Number)

   Suite 100, One Post Oak Central                            
  2000 Post Oak Boulevard, Houston, TX                        77056-4400
----------------------------------------                      ----------
(Address of Principal Executive Offices)                      (Zip Code)


       Registrant's Telephone Number, Including Area Code: (713) 296-6000



Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days.


                            YES [X]      NO [ ]






<PAGE>   2



                        PART I - FINANCIAL INFORMATION


ITEM 1 - FINANCIAL STATEMENTS


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                              STATEMENT OF INCOME
                                  (UNAUDITED)

<TABLE>
<CAPTION>
                                                     FOR THE QUARTER            FOR THE SIX MONTHS
                                                      ENDED JUNE 30,               ENDED JUNE 30,
                                                -------------------------   -------------------------
                                                   1998           1997         1998          1997
                                                -----------   -----------   -----------   -----------

<S>                                             <C>           <C>           <C>           <C>        
REVENUES:
   Oil and gas production revenues              $ 2,153,787   $ 2,699,952   $ 4,349,472   $ 6,398,643
   Interest income                                   15,437        15,770        25,248        25,162
                                                -----------   -----------   -----------   -----------

                                                  2,169,224     2,715,722     4,374,720     6,423,805
                                                -----------   -----------   -----------   -----------

EXPENSES:
   Depreciation, depletion and amortization         725,513       632,622     1,447,988     1,474,089
   Lease operating                                  184,833       166,308       390,489       257,476
   Administrative                                   140,952       135,000       275,952       270,000
   Interest                                            --            --            --          12,818
                                                -----------   -----------   -----------   -----------

                                                  1,051,298       933,930     2,114,429     2,014,383
                                                -----------   -----------   -----------   -----------

NET INCOME                                      $ 1,117,926   $ 1,781,792   $ 2,260,291   $ 4,409,422
                                                ===========   ===========   ===========   ===========

NET INCOME ALLOCATED TO:
   Managing Partner                             $   313,053   $   419,801   $   630,598   $ 1,031,257
   Investing Partners                               804,873     1,361,991     1,629,693     3,378,165
                                                -----------   -----------   -----------   -----------

                                                $ 1,117,926   $ 1,781,792   $ 2,260,291   $ 4,409,422
                                                ===========   ===========   ===========   ===========

NET INCOME PER INVESTING PARTNER UNIT           $       683   $     1,138   $     1,379   $     2,822
                                                ===========   ===========   ===========   ===========

WEIGHTED AVERAGE INVESTING PARTNER
   UNITS OUTSTANDING                                1,179.3       1,196.6       1,181.7       1,197.2
                                                ===========   ===========   ===========   ===========

</TABLE>

                The accompanying notes to financial statements
                    are an integral part of this statement.

                                       1
<PAGE>   3


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                            STATEMENT OF CASH FLOWS
                                  (UNAUDITED)

<TABLE>
<CAPTION>
                                                                         FOR THE SIX MONTHS ENDED JUNE 30,
                                                                         --------------------------------
                                                                              1998               1997
                                                                         --------------    --------------

<S>                                                                      <C>               <C>           
CASH FLOWS FROM OPERATING ACTIVITIES:
   Net income                                                            $    2,260,291    $    4,409,422
   Adjustments to reconcile net income to net cash
      provided by operating activities:
         Depreciation, depletion and amortization                             1,447,988         1,474,089
         Changes in operating assets and liabilities:
             (Increase) decrease in accrued revenues receivable                 (85,718)        1,726,230
             Decrease in accrued operating expenses payable                    (103,743)         (157,063)
             Increase (decrease) in payable to/receivable from
                 Apache Corporation                                             293,827          (933,964)
                                                                         --------------    --------------

         Net cash provided by operating activities                            3,812,645         6,518,714
                                                                         --------------    --------------

CASH FLOWS FROM INVESTING ACTIVITIES:
   Additions to oil and gas properties                                       (1,830,120)       (1,258,598)
   Non-cash portion of oil and gas property additions                          (622,547)          121,993
   Proceeds from sales of oil and gas properties                                363,534                --
   Increase in drilling advances                                                (69,012)               --
                                                                         --------------    --------------

         Net cash used in investing activities                               (2,158,145)       (1,136,605)
                                                                         --------------    --------------

CASH FLOWS FROM FINANCING ACTIVITIES:
   Repurchase of Partnership Units                                             (348,000)         (109,272)
   Distributions to Investing Partners                                               --        (1,197,827)
   Distributions to Managing Partner, net                                      (501,252)       (1,269,264)
   Payments on long-term debt                                                        --        (1,997,500)
                                                                         --------------    --------------

         Net cash used in financing activities                                 (849,252)       (4,573,863)
                                                                         --------------    --------------

NET INCREASE IN CASH AND CASH EQUIVALENTS                                       805,248           808,246

CASH AND CASH EQUIVALENTS, BEGINNING OF YEAR                                    691,797         1,737,470
                                                                         --------------    --------------

CASH AND CASH EQUIVALENTS, END OF PERIOD                                 $    1,497,045    $    2,545,716
                                                                         ==============    ==============

SUPPLEMENTAL DISCLOSURE OF CASH FLOW
   INFORMATION:
   Cash paid during the period for interest                              $           --    $       11,073
                                                                         ==============    ==============
</TABLE>

                The accompanying notes to financial statements
                    are an integral part of this statement.

                                       2
<PAGE>   4


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                                 BALANCE SHEET

<TABLE>
<CAPTION>

                                                                                JUNE 30,      DECEMBER 31,
                                                                                  1998            1997
                                                                              ------------    ------------
                                                                              (Unaudited)
                                  ASSETS

<S>                                                                           <C>             <C>         
CURRENT ASSETS:
   Cash and cash equivalents                                                  $  1,497,045    $    691,797
   Accrued revenues receivable                                                   1,623,169       1,537,451
   Receivable from Apache Corporation                                                 --           218,729
   Drilling advances                                                               141,032          72,020
                                                                              ------------    ------------

                                                                                 3,261,246       2,519,997
                                                                              ------------    ------------

OIL AND GAS PROPERTIES, on the basis of full cost accounting:
      Proved properties                                                        167,863,673     166,397,087
      Less - Accumulated depreciation, depletion
         and amortization                                                     (159,840,902)   (158,392,914)
                                                                              ------------    ------------

                                                                                 8,022,771       8,004,173
                                                                              ------------    ------------

                                                                              $ 11,284,017    $ 10,524,170
                                                                              ============    ============

                    LIABILITIES AND PARTNERS' CAPITAL

CURRENT LIABILITIES:
   Accrued exploration and development                                        $  1,142,370    $  1,764,917
   Accrued operating expenses payable and other                                      8,520         112,263
   Payable to Apache Corporation                                                    75,098            --
                                                                              ------------    ------------

                                                                                 1,225,988       1,877,180
                                                                              ------------    ------------

PARTNERS' CAPITAL:
   Managing Partner                                                                634,381         505,035
   Investing Partners (1,154.4 and 1,184.2 units
      outstanding, respectively)                                                 9,423,648       8,141,955
                                                                              ------------    ------------

                                                                                10,058,029       8,646,990
                                                                              ------------    ------------

                                                                              $ 11,284,017    $ 10,524,170
                                                                              ============    ============
</TABLE>
                The accompanying notes to financial statements
                    are an integral part of this statement.

                                       3


<PAGE>   5


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                         NOTES TO FINANCIAL STATEMENTS
                                  (UNAUDITED)


     The financial statements included herein have been prepared by the Apache
Offshore Investment Partnership (the Partnership), without audit, pursuant to
the rules and regulations of the Securities and Exchange Commission, and
reflect all adjustments which are, in the opinion of management, necessary for
a fair statement of the results for the interim periods, on a basis consistent
with the annual audited financial statements. All such adjustments are of a
normal, recurring nature. Certain information, accounting policies, and
footnote disclosures normally included in financial statements prepared in
accordance with generally accepted accounting principles have been omitted
pursuant to such rules and regulations, although the Partnership believes that
the disclosures are adequate to make the information presented not misleading.
These financial statements should be read in conjunction with the financial
statements and the summary of significant accounting policies and notes thereto
included in the Partnership's latest annual report on Form 10-K.


1.    PAYABLE TO/RECEIVABLE FROM APACHE

     Payable to/receivable from Apache Corporation, the Partnership's managing
partner (Apache or the Managing Partner), represents the net result of the
Investing Partners' revenue and expenditure transactions in the current month.
Generally, cash in this amount will be transferred to/from Apache in the month
after the Partnership's transactions are processed and the net results of
operations are determined.


2.    RIGHT OF PRESENTMENT

     In February 1994, an amendment to the Partnership Agreement created a
right of presentment under which all Investing Partners have a limited and
voluntary right to offer their Units to the Partnership twice each year to be
purchased for cash. The first right of presentment offer for 1998 was based
upon a valuation date of December 31, 1997 for a purchase price of $11,161 per
Unit, plus interest to the date of payment. The offer was made to the Investing
Partners on April 28, 1998 and, as a result, the Partnership acquired 29.832
Units for $348,000 in cash. As provided in the Partnership Agreement, as
amended (the Amended Partnership Agreement), Investing Partners will have a
second right of presentment during the fourth quarter of 1998, based on a
valuation date of June 30, 1998.

     The Partnership is not in a position to predict how many Units will be
presented for repurchase during the fourth quarter of 1998 and cannot, at this
time, determine if the Partnership will have sufficient funds available to
repurchase Units. The Partnership has no obligation to purchase any Units
presented to the extent it determines that it has insufficient funds for such
purchases. The Amended Partnership Agreement contains limitations on the number
of Units that the Partnership can repurchase, including a limit of 10 percent
of the outstanding Units on an annual basis.


                                       4

<PAGE>   6


ITEM 2 - MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND
                  RESULTS OF OPERATIONS


RESULTS OF OPERATIONS


NET INCOME AND REVENUE

     The Partnership reported net income of $1.1 million in the second quarter
of 1998, versus $1.8 million in the prior year period. Net income per Investing
Partner Unit decreased 40 percent, from $1,138 per Unit to $683 per Unit. The
decrease was attributable to lower natural gas production, lower crude oil
production and prices and higher depreciation, depletion and amortization
(DD&A) expense, mitigated by higher natural gas prices.

     For the first half of 1998, net income of $2.3 million, or $1,379 per
Investing Partner Unit, decreased 49 percent and 51 percent, respectively, from
$4.4 million, or $2,822 per Investing Partner Unit, in the same period last
year. Lower natural gas and crude oil production and prices and higher lease
operating expense (LOE) impacted 1998 results.

     Revenues decreased 20 percent, from $2.7 million in the second quarter of
1997 to $2.2 million for the same period in 1998. Natural gas and crude oil
sales contributed 83 percent and 16 percent, respectively, to the Partnership's
total revenues in the second quarter of 1998. For the first six months of 1998,
revenues decreased 32 percent to $4.4 million compared to $6.4 million for the
same period in 1997, primarily due to lower production, with natural gas and
crude oil contributing 82 percent and 17 percent, respectively, to the
Partnership's total revenues in 1998.

     The Partnership's oil and gas production volume and price information is
summarized in the following table (gas volumes presented in thousand cubic feet
(Mcf) per day):

<TABLE>
<CAPTION>
                                   FOR THE QUARTER ENDED JUNE 30,      FOR THE SIX MONTHS ENDED JUNE 30,
                                 -----------------------------------   ---------------------------------
                                                            INCREASE                            INCREASE
                                   1998         1997       (DECREASE)     1998       1997      (DECREASE)
                                 ---------   ---------     ---------   ---------   ---------   ---------
<S>                                  <C>        <C>            <C>         <C>        <C>            <C>  
Gas volume - Mcf per day             8,943      11,203         (20%)       9,010      11,398         (21%)
Average gas price - per Mcf      $    2.21   $    2.08           6%    $    2.20   $    2.46         (11%)
Oil volume - barrels per day           297         342         (13%)         315         368         (14%)
Average oil price - per barrel   $   13.13   $   18.69         (30%)   $   13.33   $   20.00         (33%)
</TABLE>


SECOND QUARTER 1998 COMPARED TO SECOND QUARTER 1997

     Natural gas sales revenues for the second quarter of 1998 totaled $1.8
million, 14 percent lower than the second quarter of 1997. The decrease
resulted from a 20 percent decline in natural gas production, negatively
impacting revenue by $.5 million. Sales volumes decreased as a result of
natural declines in production, the sale of High Island Block A-6 (Glenda
Prospect) in October 1997, and the sale of West Cameron Block 368 (Krypton
Prospect) in January 1998. Partially offsetting these decreases was a six
percent increase in natural gas prices which positively impacted revenue by $.1
million.

     The Partnership's crude oil sales revenues for the second quarter totaled
$.4 million, a 39 percent decrease from the second quarter of 1997. A 30
percent decrease in the average realized oil price negatively impacted revenues
by $.2 million.

                                       5

<PAGE>   7


YEAR-TO-DATE 1998 COMPARED TO YEAR-TO-DATE 1997

     Gas sales for the first half of 1998 of $3.6 million decreased $1.5
million, or 29 percent, when compared to the same period in 1997. Average
realized gas prices decreased $.26 per Mcf, or 11 percent, when compared with
the first six months of 1997. Gas production for the first half of 1998
decreased by 21 percent when compared to the same period in 1997, negatively
impacting revenues by $1.0 million. Production decreases in 1998 were primarily
due to natural declines in production and the sale of High Island Block A-6 and
West Cameron Block 368.

     For the six months ended June 30, 1998, oil sales decreased 43 percent to
$.8 million when compared to the same period last year. The Partnership's oil
sales revenues were negatively impacted by a 14 percent decline in oil
production and a 33 percent decrease in realized prices. The decrease in sales
volumes resulted from natural declines in production.


OPERATING EXPENSES

     The Partnership's DD&A rate, expressed as a percentage of sales, was
approximately 33 percent during the second quarter and first six months of
1998, increasing from 23 percent during the same periods in 1997. The increase
in the rate was a result of generally declining natural gas and crude oil
prices and downward reserve revisions recorded in the fourth quarter of 1997.

     Higher workover activity in the second quarter of 1998 caused LOE to be 11
percent higher when compared to the second quarter of 1997. For the first half
of 1998, LOE of $.4 million was up 52 percent when compared to the first half
of 1997. This variance was the result of higher workover activity in 1998 and
joint venture audit credits received in March 1997.


CASH FLOW, LIQUIDITY AND CAPITAL RESOURCES

CAPITAL RESOURCES AND LIQUIDITY

     The Partnership's primary capital resource is net cash provided by
operating activities, which was $3.8 million for the first half of 1998, a
decrease of 42 percent from a year ago, driven by lower oil and gas production
and prices. Future cash flows will be similarly influenced by fluctuations in
product prices, production levels and operating costs.

     The Partnership repaid all outstanding debt and terminated its revolving
credit facility on January 31, 1997, rendering the Partnership debt free for
the first time in its 14-year history. The Partnership has not incurred any new
debt in 1998.

     It is expected that net cash provided by operating activities will be
sufficient to meet the Partnership's liquidity needs through the end of 1998.
However, in the event short-term operating cash requirements are greater than
the Partnership's financial resources, the Partnership will seek short-term,
interest-bearing advances from the Managing Partner.

CAPITAL COMMITMENTS

     The Partnership's primary needs for cash are for operating expenses,
drilling and recompletion expenditures, distributions to Investing Partners,
and the purchase of Units offered by Investing Partners under the right of
presentment.

                                       6

<PAGE>   8


     During the first half of 1998, the Partnership's oil and gas property
additions totaled $1.8 million. These additions related primarily to
completions at South Timbalier Block 295, drilling activity at South Pass Block
83 and recompletions and drilling at Ship Shoal Block 259. Based on information
supplied by the operators of the properties, the Partnership anticipates
capital expenditures of approximately $2.4 million for the remainder of 1998.
The anticipated capital expenditures relate to additional recompletions at
South Timbalier Block 295 and remaining drilling activity at South Pass Block
83. Such estimates may change based on realized prices, drilling results or
changes by the operator to the development plan.

     As provided in the Amended Partnership Agreement, a first right of
presentment offer for 1998 of $11,161 per Unit, plus interest to the date of
payment, was made to Investing Partners on April 28, 1998, based on a valuation
date of December 31, 1997. As a result, the Partnership acquired 29.832 Units
for a total of $348,000 in cash. As provided in the Amended Partnership
Agreement, Investing Partners will have a second right of presentment during
the fourth quarter of 1998, based on a valuation date of June 30, 1998. The
Partnership is not in a position to predict how many Units will be presented
for repurchase during the fourth quarter and cannot, at this time, determine if
the Partnership will have sufficient funds available to repurchase Units. The
Partnership has no obligation to purchase any Units presented to the extent it
determines that it has insufficient funds for such purchases.

     Due to the reduced level of cash provided by operating activities, and the
expected need to utilize such cash to fund anticipated capital expenditures,
the Partnership did not make a distribution to Investing Partners in the first
half of 1998. The Managing Partner will periodically reassess the Partnership's
cash position to determine if a distribution will be possible in the fourth
quarter of 1998. The amount of future distributions will be dependent on actual
and expected production levels, oil and gas prices, and drilling and
recompletion expenditures.


IMPACT OF THE YEAR 2000 ISSUE

      The Year 2000 Issue poses a serious threat of business disruption to any
organization that utilizes computer technology and computer chip technology in
their business systems or equipment. Apache, as Managing Partner, manages the
Partnership's operations. Apache uses a portion of its staff for this purpose
and is reimbursed for actual costs paid on behalf of the Partnership, as well
as for general, administrative, and overhead costs properly allocable to the
Partnership. Apache has formed a Year 2000 Task Force with representation from
major business units to inventory and assess the risk of hardware, software,
telecommunications systems, office equipment, embedded chip controls and
systems, process control systems, facility control systems and dependencies on
external trading partners. A program is in place to minimize the risk of system
failure and insure critical systems are Year 2000 compliant. Apache expects
this program to be completed by mid-1999.

      The Managing Partner is in the process of replacing existing software as
it relates to Apache's accounting and financial systems to more effectively and
efficiently meet its business needs. Replacement computer systems selected by
Apache will properly recognize dates beyond December 31, 1999. The Managing
Partner presently believes that, with conversions to new software and
completion of efforts planned by the Year 2000 Task Force, the Year 2000 Issue
will be eliminated. However, if such conversions are not made, or are not
completed timely, the Year 2000 Issue could have a material impact on the
operations of the Partnership. The Managing Partner plans to replace
substantially all of its existing accounting and financial systems no later
than March 31, 1999.

      The date on which the Managing Partner plans to complete installation of
its new systems is based on management's best estimates, which were derived
using numerous assumptions of future events including the continued
availability of certain resources. However, there can be no guarantee that
these estimates will be achieved and actual results could differ materially
from those plans. Specific factors that might cause such material differences
include, but are not limited to, the availability and cost of personnel trained
in this area, and similar uncertainties.


                                       7

<PAGE>   9


FORWARD-LOOKING STATEMENTS AND RISK

     Certain statements in this report, including statements of the future
plans, objectives, and expected performance of the Partnership, are
forward-looking statements that are dependent on certain events, risks and
uncertainties that may be outside the Partnership's control and which could
cause actual results to differ materially from those anticipated. Some of these
include, but are not limited to, economic and competitive conditions, inflation
rates, legislative and regulatory changes, financial market conditions,
political and economic uncertainties of foreign governments, future business
decisions, and other uncertainties, all of which are difficult to predict.

     There are numerous uncertainties inherent in estimating quantities of
proved oil and gas reserves and in projecting future rates of production and
timing of development expenditures. The total amount or timing of actual future
production may vary significantly from reserves and production estimates. The
drilling of exploratory wells can involve significant risks, including those
related to timing, success rates and cost overruns. Lease and rig availability,
complex geology and other factors can affect these risks. Future oil and gas
prices also could affect results of operations and cash flows.

                                       8

<PAGE>   10


                          PART II - OTHER INFORMATION


ITEM 1.      LEGAL PROCEEDINGS

             None.

ITEM 2.      CHANGES IN SECURITIES

             None.

ITEM 3.      DEFAULTS UPON SENIOR SECURITIES

             None.

ITEM 4.      SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS

             None.

ITEM 5.      OTHER INFORMATION

             None.

ITEM 6.      EXHIBITS AND REPORTS ON FORM 8-K

             a.   Exhibits.

                  27.1     Financial Data Schedule.

             b. Reports on Form 8-K - None.



                                       9


<PAGE>   11



                                   SIGNATURES


Pursuant to the requirements of Section 13 or 15(d) of the Securities Exchange
Act of 1934, the registrant has duly caused this report to be signed on its
behalf by the undersigned thereunto duly authorized.


                                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                                     By:   Apache Corporation, General Partner



Dated:    August 10, 1998            /s/ Roger B. Plank
                                     ------------------------------------------
                                     Roger B. Plank
                                     Vice President and Chief Financial Officer


Dated:    August 10, 1998            /s/ Thomas L. Mitchell
                                     ------------------------------------------
                                     Thomas L. Mitchell
                                     Vice President and Controller
                                     (Chief Accounting Officer)


                                      10
<PAGE>   12
                               INDEX TO EXHIBITS


<TABLE>
<CAPTION>
EXHIBIT
NUMBER              DESCRIPTION
-------             -----------
<S>                 <C>
  27.1              Financial Data Schedule
</TABLE>