If Democrats and Republicans agree, it could be today before the US Senate signs off on a plan, close to tomorrow's deadline when the Obama administration says it will reach its borrowing limit and risk default.
Republicans in the House of Representatives were last night discussing an alternative plan that includes measures that would affect President Barack Obama's healthcare reforms, which have been at the heart of the crisis.
But Arkansas Democrat Mark Pryor said: "I think we'll get an agreement today in the Senate.
"I'm not saying we can pass it today because there's logistics about drafting and getting it to the floor but my guess is we'll pass something in the Senate tomorrow."
"There's a lot of work that's going to be done over the next two or three days," Republican Senator Bob Corker said but he cautioned it was too early to be sure of a deal.
Congressional sources said the two sides in the Senate were still at odds over Democrats' demand for a delay in an insurance fee that is part of the new healthcare reforms.
Thursday is the deadline for raising the debt ceiling to extend the government's borrowing authority. The Obama administration says it will be unable to pay all of its bills if Congress does not raise the $16.7 trillion debt limit by then.
The federal government has been in partial shutdown since October 1 when Congress missed the deadline for funding it.
Conservative Republicans prompted the crisis, which has rattled global markets, by demanding major changes to the healthcare reforms as a condition of passing a government spending bill and raising the debt ceiling.
One senior Senate aide said that, under the Senate plan being negotiated, the Treasury Department would maintain its ability to use extraordinary measures to manage debt payments if a February 2014 deadline for raising US borrowing authority is missed.
Mr Pryor and Mr Corker both said their concerns focused on fellow Republicans in the House, where conservatives who are starting to hear about the terms of the Senate agreement were already expressing scepticism, spurred on by conservative organisations.
The tentative Senate agreement does not defund or delay Obamacare, but some Senate Republicans are seeking minor changes such as tougher income verification for those seeking insurance subsidies under the law and delay to a so-called reinsurance fee.
In the last few days talks have been led by Senate Majority Leader Harry Reid, a Democrat, and his Republican counterpart, Mitch McConnell. Mr Obama telephoned McConnell on Monday evening but neither the White House nor McConnell's office gave details.
Amy Klobuchar, a Minnesota Democrat who is part of a bipartisan group of senators trying to forge a compromise, said an agreement would buy time for a more thoughtful process to settle longer-term budget issues.
The deal would reopen the federal government, pay US bills and allow the parties to commit to further negotiate, she said, adding: Those are the bright spots."
Polls show Republicans have taken a hit in opinion polls since the standoff began. A Washington Post/ABC News poll released on Monday found 74% of Americans disapprove of the way congressional Republicans have handled the standoff, compared with a 53% disapproval rating for Mr Obama.
A survey released by Gallup yesterday showed Americans' confidence in the US economy fell five points last week as the government shutdown continued.