Senate Republican leader Mitch McConnell said his party’s lawmakers are working on a bipartisan solution to the US budget stalemate.
Returning from a two-hour meeting at the White House, Mr McConnell told reporters that the session with President Barack Obama was useful and he hoped Republicans could find a way out of the impasse.
Senate Republicans have had several private discussions on how to end the partial government shutdown, now approaching its 12th day, and avert a catastrophic economic default.
Republicans are offering to pass legislation that includes cuts in benefit programmes, officials in the House of Representatives have said.
Senior aides to Speaker John Boehner and Majority Leader Eric Cantor outlined the proposal at a late-night White House meeting with senior administration officials. It would provide funding to increase the federal debt limit so the government can pay its bills on time.
The proposal includes an easing of the deep spending cuts that began taking effect earlier this year, and replacing them with curbs in benefit programmes that Mr Obama himself has backed. Among them is a plan to raise the cost of healthcare for the elderly for better-off beneficiaries.
Congress, under US law, must approve an increase to the federal debt limit so the government can pay its bills.
Normally, this is routine, but Republicans had been insisting on cuts and changes to Mr Obama’s 2010 healthcare overhaul law and other programmes as the price for reopening government and extending the debt limit past next Thursday’s deadline.
A White House statement about the meeting with House Republicans said “no specific determination was made” but added: “The President looks forward to making continued progress with members on both sides of the aisle.”