In what could be the big sleeper of the 97th Congress, a constitutional amendment requiring a three-fifths vote of both houses to unbalance future budgets is now given a fair chance of passage before the end of this year.

The amendment, which has crept up on the Capitol, is now considered likely to pass the Senate this spring and provoke a major confrontation in the House, where it has long been bottled up by the Democratic leadership.

It is being propelled by a carefully orchestrated nationwide campaign that almost daily flushes out new supporters on Capitol Hill, but which takes its force--oddly enough in this year of outcry over budget cuts--from the nation's state legislatures.

Thirty-one of the state legislatures have called for a constitutional convention to approve a balanced budget amendment. Only three more need do so to force the calling of a convention; there are about 10 states in which one legislative house has already taken the required vote, and there is only one house to go.

This confronts those in Congress who oppose the amendment as too restrictive with a choice: adopt an amendment themselves or leave the issue to a convention which could also adopt all sorts of other amendments if it chose. A convention, warns Rep. Barber B. Conable Jr. (N.Y.), ranking Republican on the House Ways and Means Committee and a House sponsor of the amendment, could become a "devil's playground" for amendments once it got going.

A balanced budget has long been a dream of conservatives who say that deficits fuel inflation and who, for reasons of social as well as economic policy, would like to see the federal government constrained.

Coincidentally, Ronald Reagan has long been a leader of those favoring a balanced budget. Yet he is now presiding over and, to some extent, insisting on the largest deficits in the nation's history, and these projected deficits themselves are adding to the pressure for an amendment.

The administration, moreover, has not announced a position on the amendment and seems to be split over it. Office of Management and Budget Director David A. Stockman has told members of Congress that he opposes it as too constraining, according to Conable, but Treasury Secretary Donald T. Regan is said to be for it.

The amendment in Congress is not the pristine, simplistic balance-the-budget commandment once advocated by tax resistance groups, but it establishes the principle that expenditures ought to match revenues over a period of time.

Congress would have to prepare for each fiscal year a budget in which outlays did not exceed revenues. In addition, these budgeted revenues could not rise from year to year by more than the growth in national income; this would limit the size of government relative to the size of the economy as a whole. Only by a vote of three-fifths of the House and the Senate could the statement be amended to permit a deficit.

Its sponsors claim the amendment is needed to break the cycle in which Congress, pressured by special-interest groups, is willing to vote separately for higher appropriations each year but unwilling to vote the large tax increases needed to finance them. As the Senate Judiciary Committee stated in its report when it sent the amendment to the floor last year, it would do away with "a fiscal order in which members of Congress have every political incentive to spend money and almost no incentive to forgo such spending."

Amendment supporters are conscious of the irony whereby the deficits of an anti-deficit president are helping their cause.

"There is no question about it," says Lewis Uhler, president of the National Tax Limitation Committee, which has promoted the issue. "The fact that a fiscally committed administration is having such difficulty with the budget underscores that there is a fundamental dysfunction in the political structure. We have a proliferation of special interests vying for their pieces of the pie and so Congress increases the size of the pie."

Conable senses that he is nearing victory because so many members know they will be voting for a huge deficit later this year. Voting as well for constitutionally balanced budgets in future years, he argues, will make the task easier.

"There are lots of Republicans bellowing about having to vote for such a big deficit this year," he said. "I'm trying to convince them they'll have to vote for a deficit in the short term but they can make that easier by voting separately for a long-term commitment to balanced budgets."

It would take a two-thirds vote of both houses to submit the amendment to the states for ratification and supporters claim they have about 60 of the needed 67 votes in the Senate; the Judiciary Committee reported it out 11 to 5. "There is a hell of a lot of steam behind it," said an aide to Majority Leader Howard H. Baker Jr. (R-Tenn.). It probably will come to the floor in late spring.

The amendment has about 175 House sponsors, but has been bottled up in the hostile House Judiciary Committee. Conable's strategy is to add an overwhelming number of sponsors, and, if necessary, force the committee to discharge the amendment.

The House and Senate Democratic leaderships have carefully avoided taking a public position but are assumed to be opposed. Sen. Max Baucus (D-Mont.) has objected to it, saying that Congress "would be locking in one rigid mechanism for restraining government spending and could frustrate the exercise of sound economic policy in the future."

Meanwhile, some Democrats in the states are also gearing up. Last Thursday the liberal Democratic minority leader of the New Hampshire House of Representatives, Chris Spirou, filed a motion to rescind that state's petition for a balanced-budget constitutional convention. Spirou said he had written Democratic leaders of other legislatures urging similar action on their part.

If proposed by Congress and approved by enough states, the amendment would take effect during the second fiscal year after ratification, probably 1986.

It is the product of a heavy lobbying campaign organized by the National Tax Limitation Committee, aided by other tax-resistance organizations which have long promoted the idea. The committee, which has about 600,000 members, has focused its grass-roots letter-writing campaign by using computers and mailing out more than 30 million pieces of direct-mail appeals in three years. It dispatched 3.2 million pieces this month, imploring members to get in touch with House members and senators.

Uhler, the committee's president, is a Loomis, Calif., businessman who served in the state cabinet when Reagan was governor and also headed Reagan's tax reduction task force.

"We are up front about what we are doing--we're using pressure and persuasion," Uhler said in a telephone interview. "When we find a senator who has not signed on, we ask his people to call him up."

The Washington office of the committee has two staff lobbyists working the Hill and has retained the services of a leading professional lobbyist, Charls E. Walker , a former deputy secretary of the Treasury.