Republican Senate leader Mitch McConnell said there was "a level of seriousness with the right people at the table" as talks continued.
Senior Senate Democrat Richard Durbin spoke of "a more positive feeling".
In a sign of the level of anxiety over the issue, troops in Afghanistan asked Adm Mike Mullen if they would be paid.
The admiral, who as chairman of the Joint Chiefs of Staff is on a visit to southern Afghanistan, said he did not know whether that would be the case if the US fails to raise the $14.3tn (?8.7tn) limit by 2 August.
Democrats and Republicans have so far rejected each others' proposals for cutting spending and raising the debt limit.
President Barack Obama backs Democratic Senate Majority Leader Harry Reid's proposal, which would cut $2.2tn from deficits and raise the debt ceiling by $2.7tn, meaning the issue would not have to be revisited until after the 2012 elections.
Late on Saturday, Mr Reid said he was postponing a planned procedural vote on his bill in the Senate - the latest stage in a series of Congressional stand-offs between the two parties.
"There are negotiations going on at the White House to avert a catastrophic default on the nation's debt. There are many elements to be finalised and there is still a distance to go," he said.
The vote, which had been expected at 0100 (0500 GMT) on Sunday, will now be held at 1300 (1700 GMT), he added.
Mr Reid's deputy, Richard Durbin, said: "We're a long way from any kind of a negotiated agreement, but there is certainly a more positive feeling about reaching an agreement this evening than I've felt in a long time."
'Reasonable people'The House of Representatives, where Republicans have a majority, already rejected the bill 246-173 on Saturday afternoon, even before the Senate had voted on it.
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Lyse Doucet BBC News, southern Afghanistan The chairman of the Joint Chiefs of Staff, Adm Mike Mullen, wanted the troops to let him know what problems they faced in this war. They were more worried about other battles - the political ones far away in Washington.
One after another, the soldiers asked: "If the government defaults on its debt, will we still get paid?" The first time it was raised, the admiral paused for a long moment then said: "I don't know the answer to that question." That raised worried eyebrows across the room.
For President Obama's top military adviser knows this is a minefield he must avoid. Tell the troops there is a plan and Republican critics back home will attack the president's doomsday scenarios if they don't reach a deal. He told his troops he understood their families were living "pay check to pay check" but added: "You should still keep showing up for work here."
Senate Democrats had hoped to pass the bill by Sunday morning so that it could go to the House by Monday.Following Saturday's House vote, President Barack Obama summoned Mr Reid and Democratic House leader Nancy Pelosi for talks.
Republicans said they were confident a deal could be reached.
Referring to the Senate bill, Senate Republican leader Mitch McConnell called on Democrats to "end this charade" so that negotiations could be pursued with the president.
"We are now fully engaged with the one person in America out of 307 million people who can sign this bill into law," he said.
The BBC's Paul Adams in Washington says the Treasury is already drawing up emergency plans in case a deal is not reached.
The US Treasury estimates that the government will no longer be able to borrow money to pay all its bills unless its borrowing limit is increased by Tuesday.
Experts say the government has enough cash to keep functioning for another week or so after that.
Republican planPoliticians have had a little breathing space over the weekend to negotiate before Asian financial markets open on Monday morning, which will be Sunday afternoon in the US capital.
Mitch McConnell: ''My view is we ought to end the charade and get serious''The House of Representatives passed its own plan on Friday evening by 218-210, with 22 Republicans and every Democrat voting against.
The plan, drawn up by House speaker John Boehner, includes some $900bn of spending cuts and would raise the debt ceiling by a similar amount.
However, it would require another vote during mid-2012 - in the midst of next year's presidential campaign - and includes language in support of a so-called "balanced budget amendment" to the US constitution. Both are rejected by the White House and the Senate leadership.
Shortly after the House passed its bill, the Democratic-led Senate voted to reject the Boehner plan.
In a weekly radio address, Mr Obama reiterated that any solution on a default had to be bipartisan.
"There are multiple ways to resolve this problem," he said.
"Congress must find common ground on a plan that can get support from both parties in the House and in the Senate. And it has to be a plan that I can sign by Tuesday."
Analysts predict a last-minute scramble for a compromise and razor-edge votes in both chambers, with the high-stakes game of legislative brinkmanship expected to continue all weekend.
The Boehner and Reid plans overlap in key ways, such as trimming spending over 10 years and shunning President Obama's call for tax increases on the wealthy and corporations.
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