Senate Republican leaders yesterday agreed to put off action on a constitutional amendment to require balanced budgets as opponents claimed enough votes to defeat the measure.

The apparent turnabout in support for the proposed amendment came as GOP leaders, who had been pushing for a vote last night, switched and moved to postpone it until March 25.

A final decision on the postponement was put off until today. But before the Senate quit for the night, GOP leaders, scrambling to pick up votes, agreed to a Democratic proposal they rejected Tuesday that would require the president to submit a balanced budget or explain why not.

Sen. Daniel J. Evans (R-Wash.), a leader of bipartisan forces opposed to the constitutional amendment, said the delay "means they party leaders don't have enough votes" to get the two-thirds majority required for approval of a constitutional amendment.

Evans said he expected the White House and other advocates of the amendment to mount a major lobbying effort in the next two weeks but he claimed "a solid 34 [votes] and maybe more than that" against the measure.

Senate rejection would constitute a major defeat for President Reagan, who has repeatedly listed a balanced-budget amendment among his top legislative priorities, and Majority Leader Robert J. Dole (R-Kan.), who has pushed hard for it.

A similar amendment was approved by the Republican Senate, 69 to 32, in 1982, but it failed to get a two-thirds majority in the Democratic-controlled House. In the meantime, Senate support appeared to be eroded by membership changes and vote switches on both sides of the aisle.

While some proponents of the measure said that enactment of the Gramm-Rudman-Hollings budget-balancing law last year would add to pressure for making the requirements permanent as part of the Constitution, Evans suggested last night that the reverse may have been true.

"People want to give Gramm-Rudman a chance to work," he said, adding that there is increasing reluctance to tamper with the Constitution before trying a legislative solution first.

The proposed constitutional amendment, designed to take effect after the Gramm-Rudman-Hollings measure produces a balanced budget in fiscal 1991, would require a three-fifths vote of the membership of both chambers to approve spending in excess of revenues. It also would require a majority of members of both chambers to approve tax increases.

As the Senate was debating the amendment, hopes for a quick bipartisan budget agreement for next year faded as Budget Committee Chairman Pete V. Domenici (R-N.M.) and the panel's ranking Democrat, Lawton Chiles (Fla.), hit a snag in private negotiations over defense spending levels. The dispute reportedly involved $14 billion in anticipated spending authority for the year and remained unresolved last night. The committee is scheduled to resume deliberations today.

Meanwhile, House Democratic leaders scheduled a vote today on Reagan's fiscal 1987 budget, prompting Republican complaints that they were engaging in "empty political shenanigans" and "president-bashing."

Reagan's budget is expected to be rejected overwhelmingly, with many Republicans voting "present," thereby avoiding a vote for or against the plan.

The Republicans' complaints prompted a countercharge from Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) that GOP members "haven't the guts to vote for it and they don't want to vote against it."

Republicans, led by Rep. Lynn M. Martin (Ill.), acting ranking minority member on the Budget Committee, accused the Democrats of foot-dragging in producing their own budget plan.

House Budget Committee Chairman William H. Gray III (D-Pa.) replied that he had asked Martin if she wanted to offer amendments or a substitute on behalf of House Republicans and that she had declined. He defended the Democrats' plan for a vote on Reagan's budget, saying the outcome would be helpful in drafting a House version.

Reagan's budget was rejected last week by the Republican-controlled Senate Budget Committee.

At a breakfast meeting with reporters yesterday, Gray reiterated that House Democrats will insist on a "nod" from Reagan before including tax increases in their budget plan. He indicated reluctance even to include a tax-amnesty plan unless the president goes along.

But Gray also voiced skepticism that Congress could reach the $144 billion fiscal 1987 deficit target imposed by Gramm-Rudman-Hollings without resorting to revenue increases.

After earlier spending cuts, "you're getting down to muscle, baby -- bone, muscle and sinew," Gray said.