WASHINGTON — A bipartisan effort in the Senate to allow President Barack Obama to raise the federal debt ceiling in exchange for about $1.5 trillion in spending cuts over 10 years gained momentum Sunday, as leaders agreed they would have to act in the next two weeks to avert a potential default by the U.S. government.

The growing sentiment for raising the federal limit on U.S. borrowing sets the stage for a week of largely scripted actions on Capitol Hill, where leaders in both chambers are looking to build support for the plan being crafted by Senate Majority Leader Harry Reid, D-Nev., and Senate Minority Leader Mitch McConnell, R-Ky.

Republican leaders will first push forward in the House and the Senate with a constitutional amendment to balance the federal budget. The measure is virtually certain to fail in the Senate, which will then take up the debt limit proposal by midweek.

If that clears the Senate, the House is expected to revise the measure, adding a proposal to reduce the deficit by $1.5 trillion over 10 years — savings that will come through cuts to domestic programs and not cuts to entitlement or new taxes.

The plan would also create a new congressional panel that would, by the end of the year, seek to come up with a way of reducing the deficit by another $2.5 trillion or more through cuts in entitlements and other steps.

Although the debt-limit plan has broad support in the Senate, the prospects in the House are less clear and rely largely on whether House Speaker John A. Boehner, R-Ohio, will bring the proposal up for a vote and how many House Democrats would support it since few Republicans are expected to get behind it.

At the same time, the White House will continue to push for as big a deal as possible to cut the deficit, which would include spending cuts and changes to entitlements as well as tax increases.

On NBC’s “Meet the Press,” Jacob Lew, Obama’s budget director, said he hoped the Republicans could compromise with Obama on a big deal. But he did not express optimism. “The question is: Do we have a partner to work with?” he asked.

Informal talks between the White House and Congress over the weekend did not appear to move the two sides significantly closer to a big deal. Leaders face an Aug. 2 deadline to raise the federal debt ceiling or face a potentially damaging government default on its obligations. They say they need to get a piece of legislation under way by week’s end to clear procedural barriers and raise the debt ceiling in time.

Most lawmakers were focusing on the new Senate plan, originally proposed last week by McConnell and further developed by Reid. Under the plan, Obama would be able to raise the debt ceiling three times in the next year for a total of $2.5 trillion.

Congress could also vote on a resolution of disapproval each time, assigning blame to Obama for increasing the nation’s debt.

In addition to the $1.5 trillion in spending cuts, the plan would create a new committee of 12 lawmakers, which would issue a report to Congress by the end of the year on how to cut $2.5 trillion or more from federal deficits in the next 10 years.

This committee would seek agreement where Obama and Republicans haven’t been able — primarily, over changes to entitlement programs and whether raising new tax revenue should play an important role in cutting the deficit.

Only a deal to cut the deficit by $4 trillion or more would make the debt manageable over time. The credit rating company Standard & Poor’s said last week that any deal short of that could lead to a downgrade of the nation’s credit rating.

Such a deal could still be achieved in the coming weeks, though the prospects have dimmed. Obama and Boehner have both shown a keen interest in a “grand bargain,” but the issue of taxes has so far proven intractable.

Before taking up the McConnell-Reid plan, the House and Senate will consider the “cut, cap and balance” approach being pushed by Republicans. The House is slated to vote early this week on a balanced budget amendment to the Constitution that would significantly scale back government spending and make it harder for lawmakers to raise taxes.

Nearly 40 House Republicans have said they would not support an increase in the debt ceiling without such a constitutional amendment. But Democrats strongly oppose the measure.