<PAGE>   1
                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                    FORM 10-Q

(MARK ONE)

[ X ]   QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
        OF THE SECURITIES EXCHANGE ACT OF 1934 FOR THE QUARTERLY
        PERIOD ENDED SEPTEMBER 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 0-16614

                                NEORX CORPORATION
             (Exact Name of Registrant as Specified in its Charter)

<TABLE>
<S>                                                    <C>       
           WASHINGTON                                      91-1261311
(State or other jurisdiction of                          (IRS Employer
 incorporation or organization)                        Identification No.)
</TABLE>

               410 West Harrison Street, Seattle, Washington 98119
             (Address of Principal Executive Offices)     (Zip Code)

       Registrant's telephone number, including area code: (206) 281-7001



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   
                                         -----      -----
Applicable only to corporate issuers:

Indicate the number of shares outstanding of each of the issuer's classes of
common stock as of the latest practicable date.

As of October 9, 1998 there were outstanding 21,006,964 shares of the Company's
Common Stock, $.02 par value.


<PAGE>   2



                                TABLE OF CONTENTS

                          QUARTERLY REPORT ON FORM 10-Q
                    FOR THE QUARTER ENDED SEPTEMBER 30, 1998


<TABLE>
<CAPTION>
PART I                       FINANCIAL INFORMATION                    PAGE
<S>            <C>                                                    <C> 
Item 1.         Financial Statements:

                Balance Sheets as of September 30, 1998
                and December 31, 1997                                   3

                Statements of Operations for the
                three and nine months ended September 30,
                1998 and 1997                                           4

                Statements of Cash Flows for the
                three and nine months ended September 30,
                1998 and 1997                                           5

                Notes to Financial Statements                           6

Item 2.         Management's Discussion and Analysis
                of Results of Operations and
                Financial Condition                                     9

Item 5.         Other Information                                       11

                Signature                                               12
</TABLE>



                                       2
<PAGE>   3

                                NEORX CORPORATION

<TABLE>
<CAPTION>
BALANCE SHEETS
(in thousands, except share data)
                                                                                                SEPTEMBER 30,           DECEMBER 31,
                                                                                                    1998                    1997
                                                                                                -------------           ------------
                                                                                                (unaudited)
<S>                                                                                             <C>                     <C>
                                     ASSETS
CURRENT ASSETS:
  Cash and cash equivalents                                                                       $   1,798               $   1,949
  Short-term investments                                                                             30,786                  31,760
  Prepaids and other                                                                                    882                   1,820
                                                                                                  ---------               ---------
    Total current assets                                                                             33,466                  35,529
                                                                                                  ---------               ---------
FACILITIES AND EQUIPMENT, at cost:
  Leasehold improvements                                                                              3,260                   3,300
  Equipment and furniture                                                                             4,665                   4,023
                                                                                                  ---------               ---------
                                                                                                      7,925                   7,323
  Less: accumulated depreciation and amortization                                                    (6,890)                 (6,642)
                                                                                                  ---------               ---------
    Facilities and equipment, net                                                                     1,035                     681
                                                                                                  ---------               ---------
OTHER ASSETS                                                                                            472                     111
                                                                                                  ---------               ---------
    Total assets                                                                                  $  34,973               $  36,321
                                                                                                  =========               =========


                      LIABILITIES AND SHAREHOLDERS' EQUITY

CURRENT LIABILITIES:
  Accounts payable                                                                                $     760               $     800
  Accrued liabilities                                                                                   855                     911
  Current portion of capital leases                                                                      15                      43
                                                                                                  ---------               ---------
    Total current liabilities                                                                         1,630                   1,754
LONG-TERM LIABILITIES:
  Convertible subordinated debentures, 9 3/4%                                                         1,195                   1,195
  Capital leases, less current portion                                                                   --                       4
                                                                                                  ---------               ---------
    Total liabilities                                                                                 2,825                   2,953
                                                                                                  ---------               ---------

COMMITMENTS AND CONTINGENCIES

SHAREHOLDERS' EQUITY:
  Series preferred stock, $.02 par value,
    3,000,000 shares authorized:
      Convertible exchangeable preferred stock, Series 1,
      208,240 shares issued and outstanding at September 30,
      1998 and December 31, 1997 (entitled in liquidation
      to $5,248 at September 30, 1998 and December 31, 1997)
    Convertible preferred stock, Series 2, -0- shares
      issued and outstanding at September 30, 1998 and 5,167
      shares issued and outstanding at December 31, 1997
    Convertible preferred stock, Series 3, -0- shares
      issued and outstanding at September 30, 1998 and 1,000
      shares issued and outstanding at December 31, 1997                                                  4                       4
  Common stock, $.02 par value, 60,000,000 shares authorized,
    21,006,964 and 20,707,251 shares issued and outstanding
    at September 30, 1998 and December 31, 1997, respectively                                           420                     414
  Additional paid-in capital                                                                        163,189                 162,612
  Accumulated deficit                                                                              (131,465)               (129,662)
                                                                                                  ---------               ---------
    Total shareholders' equity                                                                       32,148                  33,368
                                                                                                  ---------               ---------
Total liabilities and shareholders' equity                                                        $  34,973               $  36,321
                                                                                                  =========               =========
</TABLE>


               See accompanying notes to the financial statements.


                                       3
<PAGE>   4

                                NEORX CORPORATION




STATEMENTS OF OPERATIONS
(In thousands, except per share data)
(Unaudited)




<TABLE>
<CAPTION>
                                        THREE MONTHS               NINE MONTHS
                                      ENDED SEPTEMBER 30,       ENDED SEPTEMBER 30,
                                    ---------------------     ---------------------
                                      1998          1997        1998          1997
                                    --------     --------     --------     --------
<S>                                 <C>          <C>          <C>          <C>     
REVENUES                            $    149     $  5,002     $  7,697     $ 10,349
                                    --------     --------     --------     --------

OPERATING EXPENSES:

  Research and development             2,648        2,804        7,128        8,194

  General and administrative             953          938        3,343        2,912
                                    --------     --------     --------     --------

    Total operating expenses           3,601        3,742       10,471       11,106
                                    --------     --------     --------     --------

Income (loss) from operations         (3,452)       1,260       (2,774)        (757)

OTHER INCOME (EXPENSE):
  Investment and interest income         358          507        1,467        1,289

  Interest expense                       (32)         (33)         (98)        (102)
                                    --------     --------     --------     --------

Net income (loss)                   $ (3,126)    $  1,734     $ (1,405)    $    430
                                    ========     ========     ========     ========

Preferred stock dividends               (127)        (315)        (400)      (2,923)
                                    --------     --------     --------     --------

Net income (loss) applicable to
 Common shares                      $ (3,253)    $  1,419     $ (1,805)    $ (2,493)
                                    ========     ========     ========     ========

Earnings (loss) per share:

Basic                               $   (.15)    $    .08     $   (.09)    $   (.15)
                                    ========     ========     ========     ========

Diluted                             $   (.15)    $    .07     $   (.09)    $   (.15)
                                    ========     ========     ========     ========

Shares used in calculation of
Earnings (loss) per share:

Basic                                 21,004       18,288       20,874       17,193
                                    ========     ========     ========     ========

Diluted                               21,004       19,015       20,874       17,193
                                    ========     ========     ========     ========
</TABLE>


               See accompanying notes to the financial statements.


                                       4
<PAGE>   5

                                NEORX CORPORATION

STATEMENTS OF CASH FLOWS
(In thousands)
(Unaudited)
<TABLE>
<CAPTION>    
                                                THREE MONTHS                NINE MONTHS
                                              ENDED SEPTEMBER 30,       ENDED SEPTEMBER 30,
                                              -------------------       -------------------
                                               1998         1997          1998        1997
                                               ----         ----          ----        ----
<S>                                         <C>           <C>          <C>         <C>
CASH FLOWS FROM OPERATING ACTIVITIES:

Net income (loss)                            $ (3,126)    $  1,734     $ (1,405)    $    430
Adjustments to reconcile net income
  (loss)to net cash provided by
  (used in) operating activities:
    Depreciation and amortization                  90           76          265          272
    (Increase) decrease in prepaids
      and other assets                            393         (285)         542         (345)
    (Decrease) in accounts payable                  0         (118)         (40)        (335)
    (Decrease) in accrued liabilities              19           26          (97)        (127)
    Common stock issued for services                           101            -          101


Net cash provided by (used in) operating
  activities                                   (2,624)       1,534         (735)          (4)
                                             --------     --------     --------     --------

CASH FLOWS FROM INVESTING ACTIVITIES:

Proceeds from sales of short-
  term investments                             23,274       12,419       57,095       37,984
Purchases of short-term investments           (20,377)     (17,849)     (56,122)     (54,743)
Facilities and equipment purchases                (75)           -         (602)        (261)
Other                                               7            5           19           17
                                             --------     --------     --------     --------
Net cash provided by (used in)
  investing activities                          2,829       (5,425)         390      (17,003)
                                             --------     --------     --------     --------

CASH FLOWS FROM FINANCING ACTIVITIES:

Proceeds from sale of common stock                  -            -            -        2,661
Proceeds from the sale of preferred stock           -        4,910            -       16,370
Repayments of capital lease obligations           (11)          (9)         (32)         (36)
Proceeds from stock options and warrants
 exercised                                         35          162          482          297
Preferred stock dividends                           0         (130)        (256)        (385)
                                             --------     --------     --------     --------
Net cash provided by financing
  activities                                       24        4,933          194       18,907
                                             --------     --------     --------     --------

NET INCREASE (DECREASE) IN CASH
  AND CASH EQUIVALENTS
                                                  229        1,042         (151)       1,900
CASH AND CASH EQUIVALENTS:
Beginning of period                             1,569        3,803        1,949        2,945
                                             --------     --------     --------     --------

End of period                                $  1,798     $  4,845     $  1,798     $  4,845
                                             ========     ========     ========     ========
</TABLE>



               See accompanying notes to the financial statements.



                                       5
<PAGE>   6

                                NEORX CORPORATION


NOTES TO FINANCIAL STATEMENTS

Note 1. Basis of Presentation

The interim financial statements contained herein have been prepared pursuant to
the rules and regulations of the Securities and Exchange Commission. Certain
information and note disclosures normally included in annual financial
statements prepared in accordance with generally accepted accounting principles
have been condensed or omitted pursuant to those rules and regulations, although
the Company believes that the disclosures made are adequate to make the
information presented not misleading. These financial statements should be read
in conjunction with the Company's annual report on Form 10-K for the year ended
December 31, 1997.

Certain reclassifications were made to the 1997 financial statements to make
them comparable with the 1998 presentation.

In the opinion of management, the interim financial statements reflect all
adjustments, consisting only of normal recurring accruals necessary to present
fairly the Company's financial position as of September 30, 1998 and the results
of operations and cash flows for the three and nine month periods ended
September 30, 1998 and 1997.

The results of operations for the three and nine month periods ended September
30, 1998 are not necessarily indicative of the expected operating results for
the full year.

Note 2. Shareholders' Equity

Changes in shareholders' equity from December 31, 1997 to September 30, 1998
were as follows (in thousands):

<TABLE>
<S>                                                  <C>     
                 Balance December 31, 1997           $ 33,368
                 Common stock issued                      585
                 Preferred stock dividends               (400)
                 Net loss                              (1,405)
                                                     --------

                 Balance September 30, 1998          $ 32,148
                                                     ========
</TABLE>

                                       6
<PAGE>   7

                                NEORX CORPORATION


NOTES TO FINANCIAL STATEMENTS (Continued)


Note 3. Earnings (loss) per share:

The following is a reconciliation of the numerator and denominator of the basic
and diluted earnings (loss) per share computations for the three months and nine
months ended September 30, 1997 and 1998:

<TABLE>
<CAPTION>
                                Three months              Nine months
                            ended September 30,       ended September 30,
                            -------------------       -------------------
                              1998         1997         1998         1997
                              ----         ----         ----         ----
<S>                        <C>         <C>           <C>          <C>

Net Income (loss)          $ (3,126)    $  1,734     $ (1,405)    $    430
Less: Preferred
 stock dividends               (127)        (315)        (400)      (2,923)
                           --------     --------     --------     --------
Net Income (loss)
 applicable to basic
 & diluted earnings
 (loss) per share          $ (3,253)    $  1,419     $ (1,805)    $ (2,493)
                           ========     ========     ========     ========

Weighted average
 shares - basic              21,004       18,288       20,874       17,193
Dilutive effect of
 options & warrants               -          727            -            -
                           --------     --------     --------     --------
Weighted average
 shares - diluted            21,004       19,015       20,874       17,193
                           ========     ========     ========     ========

Basic earnings (loss)
 per share                 $   (.15)    $    .08     $   (.09)    $   (.15)
                           ========     ========     ========     ========
Diluted earnings (loss)
 per share                 $   (.15)    $    .07     $   (.09)    $   (.15)
                           ========     ========     ========     ========
</TABLE>


The numerator and denominator of the basic and diluted loss per share
calculations for the three months ended September 30, 1998 were the same,
because including the effect of options to purchase an additional 1,289,859
shares of Common Stock would have been antidilutive. Options to purchase 955,115
additional shares of Common Stock were outstanding during the three months ended
September 30, 1997 but were not included in the computation of diluted earnings
per share because the options' average exercise price of $8.53 was greater than
the average market price of the common shares of $5.33. Warrants to purchase an
additional 1,033,727 shares of Common Stock were outstanding during the three
months ended September 30, 1997 but were also not included in the computation of
diluted earnings per share because the warrants' average exercise price of
$13.63 was greater than the average market price of the common shares of $5.33.
Options to purchase 1,519,845 additional shares of Common Stock were outstanding
during the nine months ended September 30, 1998 but were not included in the
computation of diluted earnings per share because the effect of including such
shares would have been antidilutive.


                                       7
<PAGE>   8
                                NEORX CORPORATION

NOTES TO FINANCIAL STATEMENTS (Continued)

The numerator and denominator of the basic and diluted loss per share
calculations for the nine months ended September 30, 1997 were the same, because
including the effect of options and warrants to purchase an additional 1,771,160
shares of Common Stock would have been antidilutive.

Shares issuable upon conversion of the Company's Convertible Subordinated
Debentures, Series 1 Preferred Stock, Series 2 Preferred Stock and Series 3
Preferred Stock are not included in the calculation of diluted EPS for the three
and nine month periods ended September 30, 1998 and 1997 because the effect of
including such shares would have been antidilutive.


                                       8
<PAGE>   9
                                NEORX CORPORATION

Item 2. Management's Discussion and Analysis of Results of Operations and
        Financial Condition

This discussion contains forward-looking statements that are subject to certain
risks and uncertainties that could cause actual results to differ materially
from those projected. The words "believe", "expect", "intend", "anticipate", and
variations of such words and similar expressions identify forward-looking
statements, but their absence does not mean the statement is not forward
looking. Factors that could affect the Company's actual results include, among
other things, results of research and development activities, clinical trials,
expenses associated with expanding marketing and manufacturing capabilities,
competitive and technological developments, and the timing, cost and successful
continuation of the Company's collaborative relationships. Reference is made to
the Company's Annual Report on Form 10-K filed with the Commission for a more
detailed description of such factors. Readers are cautioned not to place undue
reliance on these forward-looking statements, which speak only as of the date of
this report. The Company undertakes no obligation to update publicly any
forward-looking statement to reflect new information, events or circumstances
after the date of this report or to reflect the occurrence of unanticipated
events.


Quarter and nine months ended September 30, 1998 compared to quarter and nine
months ended September 30, 1997.

Revenues for the three months ended September 30, 1998 were $149,000 compared to
$5,002,000 for the three months ended September 30, 1997. In the third quarter
1997, NeoRx entered into an agreement with Janssen Pharmaceutica, ("Janssen")
N.V., a wholly-owned subsidiary of Johnson & Johnson (NYSE: JNJ), for the
worldwide development, manufacture and distribution of NeoRx's Avicidin(R)
cancer therapy product. NeoRx received $10 million up front; $5 million was for
the purchase of Series 4 Convertible Preferred Stock and the remaining $5
million was recorded as revenue. In September 1998, Janssen terminated this
agreement effective December 29, 1998. After this date the Company will no
longer receive milestone payments or expenses reimbursements, from Janssen.

Revenues for the nine months ended September 30, 1998 were $7,697,000 compared
to $10,349,000 for the nine months ended September 30, 1997. In January 1998,
the Company received a $7,000,000 milestone payment from Janssen, reflecting
Janssen's decision to begin Phase II trials of Avicidin(R) cancer therapy
product. 1997 revenues consist of Janssen licensing fees of $5 million for
Avicidin(R) cancer therapy product and $5.3 million in license fees from Schwarz
Pharma AG covering NeoRx's Biostent(R) product.


                                       9
<PAGE>   10
                                NEORX CORPORATION

Total operating expenses for the quarter ended September 30, 1998 decreased 4%
to $3,601,000 from $3,742,000 in the quarter September 30, 1997, and for the
nine month period decreased 6% to $10,471,000 from $11,106,000. Research and
development expenses for the quarter ended September 30, 1998 decreased 6% to
$2,648,000 from $2,804,000 for the same time period in 1997. Research and
development expenses for the nine months ended September 30, 1998 decreased 13%
to $7,128,000 from $8,194,000. The decrease in research and development expenses
for both the quarter and the year is the result of decreased clinical trial
costs and manufacturing costs.

The Company receives expense reimbursements related to the Avicidin(R) and
Biostent(R) programs. Reimbursed research and development expenses totaled
$1,801,000 for the nine months ended September 30, 1998. In the first nine
months of 1997, the Company's research and development expense reimbursements
totaled $2,137,000.

General and administrative expenses decreased 30% from the previous quarter and
were essentially unchanged from the same quarter in 1997. The decline from the
second quarter in 1998 was due to lower costs from outside consulting services
and general legal services. General and administrative expenses for the nine
months ended September 30, 1998 increased 15% to $3,343,000 from $2,912,000.
General and administrative expenses for the nine months ended September 30, 1998
increased principally due to increased expenses for personnel and outside
consulting services. These costs were added to support increased research and
product development activities.

Investment and interest income for the quarter decreased to $358,000 from
$507,000 for the same time period in 1997, and increased to $1,467,000 from
$1,289,000 for the nine months ended September 30, 1998 and 1997, respectively.
The decrease in interest income for the current quarter is due to the decrease
in funds available for investment. The increase in interest income for the nine
months ended September 30, 1998 is due to higher average cash balances, which
resulted from milestone payments and equity proceeds.

Liquidity and capital resources.

Cash and short-term investments as of September 30, 1998 were $32,584,000
compared to $33,709,000 at December 31, 1997. The third quarter balance of cash
and short-term investments decreased from the 1998 year-to-date net loss.

The Company expects that its capital resources and interest income will be
sufficient to finance its currently anticipated working capital and capital
requirements at least through the year 1999.

The Company's working capital and capital requirements will depend 


                                       10
<PAGE>   11
                                NEORX CORPORATION

upon numerous factors, including results of research and development activities,
clinical trials, expenses associated with expanding marketing and manufacturing
capabilities, competitive and technological developments and the timing, cost
and successful continuation of the Company's collaborative relationships. The
Company will need to raise substantial additional funds to conduct research and
development activities, preclinical studies and clinical trials necessary to
bring its potential products to market, and to establish marketing and
manufacturing capabilities. The Company intends to seek additional funding
through public or private equity financing, arrangements with corporate
collaborators, out-licensing certain technologies, or other sources.

Adequate funds may not be available when needed or on terms acceptable to the
Company. If funding is insufficient at any time in the future, the Company will
be forced to delay, reduce or eliminate some or all of its research and
development activities, clinical studies and trials, and manufacturing and
administrative programs, dispose of assets or technology, or cease operations.

Impact of the year 2000

Overview

The Year 2000 Problem is pervasive and complex, as virtually every computer 
operation will be affected in some way by the rollover of the two-digit year 
value to 00. The issue is whether computer systems will properly recognize date 
sensitive information when the year changes to 2000. Systems that do not 
properly recognize date sensitive information could generate erroneous data or 
cause a system to fail.

Readiness

The Company has undertaken an initial review of its information technology 
computer systems and believes that the Year 2000 problem does not pose 
significant operational problems to its information technology systems. The 
majority of Company's software and computer equipment has been purchased within 
the last five years from third-party vendors who have already provided upgrades 
intended to bring their products into Year 2000 compliance. The Company plans 
to survey its research and development equipment and significant suppliers, 
including Clinical Research Organizations, to determine any additional needed 
Year 2000 solutions.

Costs

At this time, the Company cannot determine the full cost of correcting any
potential year 2000 problems. Based upon the Company's initial review of its
computer systems, the Company estimates that the cost to replace older,
non-compliant computers and software is immaterial. The full estimated cost of
correcting the Year 2000 issue will be known after the Company completes its
survey of its research and development equipment and significant suppliers.

Risks

The Company anticipates that its largest Year 2000 risks are in activities 
involving research and development equipment and significant suppliers, 
including Clinical Research Organizations. If Year 2000 problems exist in these 
areas, it could take significantly longer for the Company to bring a product to 
market, which could have a material adverse effect on the Company's business, 
financial condition and results of operations.

Contingency

After the Company completes its survey of Year 2000 issues, it will establish a 
contingency plan to address any "high-risk" issues that could delay its efforts 
to bring products to market.

Item 5. Other Information

In November 1998, the Company licensed its Paclitaxel technology for
non-oncological applications to Angiotech Pharmaceuticals, Inc., ("Angiotech").
Under the terms of the agreement, Angiotech will pay NeoRx Corporation up front
license fees of $500,000 in cash and $500,000 in equity of Angiotech.  The
agreement also includes milestones, warrants and other payments.

                                       11
<PAGE>   12
                                NEORX CORPORATION

                                    SIGNATURE
                                    ---------

Pursuant to the requirements 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.

                                            NeoRx Corporation
                                            (Registrant)



Date: November 4, 1998                       By: /s/ RICHARD L. ANDERSON
                                               --------------------------
                                               Richard L. Anderson
                                               Senior Vice President,
                                               Finance and Operations,
                                               (Principal Financial and
                                               Accounting Officer)

                                       12