Senate Republican leader Mitch McConnell has said Republican lawmakers are working on trying to come up with a bipartisan solution to the 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 that Senate 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 in its 11th day, and avert a catastrophic economic default.
Mr Obama held discussions with Senate Republicans at the White House this morning.
Republicans are offering to pass legislation to avert a debt default and end the partial government shutdown as part of a package 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 end the government shutdown and avert a potentially catastrophic default by increasing the federal debt limit so the government can pay its bills on time.
The proposal includes an easing of the deep across-the-board 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 health care for the elderly for better-off beneficiaries.
Mr Boehner brought the proposal to yesterday's White House meeting to extend federal borrowing authority and avoid default until November 22, conditioned on Mr Obama's agreeing to negotiate over spending cuts and the government shutdown. But participants said the discussion expanded to ways to quickly end the federal government shutdown.
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 health care overhaul law and other programmes as the price for reopening government and extending the debt limit past next Thursday's deadline.
Mr Boehner's offer indicates some movement away from that strategy, but Republicans claimed victory on one front, noting that they were in negotiations with a president who had said repeatedly there would be none until the government was open and default prevented.
A White House statement about yesterday's 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."
The White House said Mr Obama welcomes a "constructive approach" from congressional Republicans in a new budget offering but "has some concerns with it".
White House press secretary Jay Carney said Mr Obama and Mr Boehner spoke by phone but he would not get into details about the call. But he reiterated that Mr Obama believes the debt ceiling should be lifted without ties to budget negotiations.
Mr Carney said a short-term funding bill and debt limit increase are "the very least that Congress could do".