They have backed off from demands that any immediate extension of the government's borrowing authority be accompanied by stiff spending cuts.
But the retreat came with a caveat aimed at prodding Senate Democrats to pass a budget after almost four years of failing to do so - a threat to cut off the pay of politicians in either the House or Senate if their chamber fails to pass a budget this year. House Republicans have passed budgets for two consecutive years.
The idea received a frosty reception from House Democrats, but a more measured response from the White House and Democratic Senate majority leader Harry Reid.
Republicans had not settled on full details, but the measure would give the government about three more months of borrowing authority beyond a deadline expected to hit as early as mid-February, number two House Republican Eric Cantor said.
The legislation would not require immediate spending cuts as promised earlier by Republican leaders like House speaker John Boehner. Instead, it is aimed at forcing the Democratic-controlled Senate to join the House in debating the budget. "We are going to pursue strategies that will obligate the Senate to finally join the House in confronting the government's spending problem," Mr Boehner told Republican politicians at a retreat in Williamsburg, Virginia. "The principle is simple - 'no budget, no pay'."
Republican leaders have been grappling with how to gain leverage in their battles with Mr Obama over the budget. Mr Boehner successfully won about two trillion dollars in spending cuts as a condition of increasing the government's borrowing cap in 2011.
Mr Obama, however, was dealt a stronger hand by his re-election in November and successfully pressed through a 10-year, 600 billion-dollar increase on upper-bracket taxpayers earlier this month.
Other choke points remain, including sharp, across-the-board spending cuts that would start to strike the Pentagon and domestic programmes alike on March 1 and the possibility of a partial government shutdown with the expiration of a temporary budget measure on March 27.
Failing to meet those deadlines would have far less serious consequences than defaulting on US obligations like payments to bondholders, recipients of Social Security pension payments and myriad other commitments when the government confronts a cash crisis and can no longer borrow to make payments. That could cause a meltdown in financial markets and would inflame voters already disgusted with Congress.