Washington — The proposed constitutional amendment to balance the budget is quickly sinking into the quicksand of the legislative process. Unless backers can pull off some difficult rescue maneuvers, the amendment will not pass the 97th Congress.
And once the political pressures of the November elections have abated, critics claim, many congressmen would be perfectly happy to see the measure disappear.
''What happens after the elections?'' chortles a Democratic congressional aide. ''Conventional wisdom is that this whole thing would go away.''
''We're not giving up hope,'' counters John Buckley, legislative director of the pro-amendment National Tax Limitation Committee.
On Monday supporters of the balanced-budget amendment missed a crucial deadline for getting the measure through the House.
The amendment is currently stuck in the House Judiciary Committee, whose chairman, Rep. Peter Rodino (D) of New Jersey, is a staunch opponent. To pry the constitutional curb loose from Mr. Rodino's grasp, Republican representatives filed a discharge petition that would force the issue to a vote on the floor.
Under complex House rules, supporters needed 218 signatures by Monday to succeed. Only 201 congressmen signed the petition by the deadline.
Backers can still garner signatures for their petition - but once they reach the required number, House Speaker Thomas P. O'Neill Jr. can postpone action for seven legislative days. After that, a vote on the discharged amendment can take place only on the second or fourth Monday of a month.
These criteria mean that between now and early October, when the House will adjourn for the year, there are only two days when the balanced-budget amendment could be brought up for a vote. And, by a quirk of the calender, both are holidays: Sept. 27, Yom Kippur; and and Oct. 11, Columbus Day. The House won't be in session.
Without some special dispensation, the House won't consider the measure during the 97th Congress. Senate passage will have been for naught. Backers will have to start all over again with the 98th Congress.
''There's a timing problem here,'' admits an aide to a Republican congressmen who supports the measure.
Proponents of the balanced-budget amendment say they are still pushing for Congress to pass the measure this session. In particular, they mention two methods by which the amendment may yet be rescued:
1. Get President Reagan to call a special, postelection session of Congress, during which the amendment could be considered. This is a distinct possibility, as another controversial issue - social security - may need similar treatment.
2. Amend the rules governing discharge petitions. Proponents grouse that they have been boxed in by the calendar, and that House rules should allow some flexibility when holidays interfere with considering a discharge petition. They should be allowed to bring the matter up on the Tuesday following a holiday, they say.
But both these tactics depend on the discharge petition's eventually garnering 218 signatures - and the number of willing congressmen with pens has dropped off sharply since soon after President Reagan's public pitch for the amendment.
''I understand the number of signatures has been hovering some,'' a Republican congressional aide says.
And some Democrats say that once the elections are over much of the amendment's political support will dwindle away. The sudden visibility of the amendment, they say, was due to its usefulness as a political fig leaf, covering embarrassment over the prospect of $100 billion-plus deficits.
Once the acute need for congressmen to impress voters as being fiscally responsible has passed, amendment critics say, congressional doubts about the measure's viability will resurface.
''The lack of pressure of an election year might make a difference,'' admits Mr. Buckley of the National Tax Limitation Committee. But he adds that the American public wants this amendment, that if it came up before the full House, it would pass, and that the NTLC will ''continue to focus a spotlight on the problem, which is Peter Rodino and Tip O'Neill.''
Representative Rodino is, in fact, beginning to move on his own version of the amendment. His subcommittee on monopolies and commercial law began drawing up such a measure Wednesday. But, according to a spokesman for Mr. Rodino, progress will be ''deliberate.''
And backers of the amendment have already written off Rodino's effort, saying the Judiciary Committee is unlikely to produce anything acceptable to them.
Backers do have a final card to play. If Congress fails them, they can always redouble efforts to persuade states to call a constitutional convention for consideration of the subject. Thirty-one of the necessary 34 state legislatures have already passed resolutions calling for such a convention. Since January, three others - Washington, Missouri, and Kentucky - have passed a convention call in one legislative chamber.