Senate Majority Leader Harry Reid, D-Nev., and Minority Leader Mitch McConnell, R-Ky., employed shuttled diplomacy of sorts, using aides to run proposals back and forth though their offices are located just 50 paces apart, The Washington Post reported.
The Post said Reid was monitoring the negotiations via telephone and McConnell left the Capitol shortly before 7 p.m.
"We've been trading paper all day and talks continue into the evening," McConnell said.
The Post said Reid and McConnell have set a deadline of about 3 p.m. Sunday for wrapping up a deal.
President Barack Obama and congressional Democrats would like to bring in more revenue by raising income tax rates for income above $250,000, while preserving current rates for all income under $250,000. Republicans apparently hope to increase the limit to $400,000 and would also like to keep estate taxes low.
In his weekly radio talk, Obama urged Congress, if no deal is in place, to pass legislation extending tax cuts for those with incomes under $250,000, and to extend long-term unemployment insurance benefits.
If no budget deal is in place by Tuesday, the tax cuts passed under President George W. Bush would end and automatic across-the-board cuts in federal spending would kick in -- creating what has come to be called the fiscal cliff.
House Republican leaders said they were considering calling a hearing for Rules Committee Sunday to establish rules for a possible vote on a bipartisan deal.
Rules Committee Chairman David Dreier of California told Roll Call, "We've been working, and we're ready to go as soon as we come up with something."
The budget deal is likely to include many issues besides tax rates, including the alternative minimum tax, and House Republican leaders have notified members votes may be held through Jan. 3, Roll Call reported.
Dreier said House Speaker John Boehner, R-Ohio, and other Republicans officials would need time to review any agreement Reid and McConnell are able to get through the Senate.
"There is no agreement worthy of the American people to be found in the current Washington charade. Better if everyone went home immediately," the Post reported Gingrich said in a statement Saturday.
Likewise, Maya MacGuineas, president of the bipartisan Committee for a Responsible Federal Budget who is pushing the Fix the Debt campaign, said a narrow agreement would miss the big deficit picture.
"A small deal solves virtually nothing," she said. "We still have to find ways to cut spending, reform entitlements, raise more revenues and get the debt under control."