At stake in the negotiations, according to a number of economists, is the fate of a still fragile U.S. economy that could be pushed back into a recession by the broad tax hikes and automatic $110 billion cuts to domestic and military spending spelled out by the "fiscal cliff" legislation.
Aides for Senate Majority Leader Harry Reid, D-Nevada, and Minority Leader Mitch McConnell, R-Kentucky, said no details on the negotiations were expected until at least early afternoon when the Senate convenes a session at 1 p.m. ET.
"We've been trading paper all day, and the talks continue into the evening," McConnell told reporters Saturday night. "We've been in discussions all day. We'll let you know as soon as we have some news to make."
Even so, it was unknown if Reid and McConnell could come up with a deal that would be acceptable to House Republicans, who refused just before Christmas to take up a compromise bill because it raised taxes and had no chance of passing the Senate.
President Barack Obama was widely expected during his scheduled appearance Sunday on NBC's "Meet the Press" to call for an immediate vote by Congress on a scaled back plan that would only extend middle class tax breaks and unemployment benefits, if the congressional negotiations fail.
The president's appearance on a political talk show is his first in three years, and clearly appears timed to put pressure on lawmakers to get a deal done or take a vote.
The principal dispute continues to be over taxes, specifically over Democrats' demand to extend tax cuts passed under President George W. Bush for families making less than $250,000 a year, while raising the rates on those making more than that.
The expectation is Republicans will try to raise that income threshold to $400,000 and push to keep estate taxes low; Democrats have said they might be open to one such scenario, but not both.
Obama and Democrats have leverage, based on the president's re-election last month and Democrats' gains in the House and Senate in the new Congress. In addition, polls consistently show majority support for Obama's position on taxes, and Democrats insist the House would pass the president's plan with Democrats joined by some Republicans if House Speaker John Boehner, R-Ohio, allowed a vote on it.
"We're now at a point where, in just a couple of days, the law says that every American's tax rates are going up. Every American's paycheck will get a lot smaller, and that would be the wrong thing for our economy," the president said in his weekly address broadcast Saturday.
On Friday, following a meeting with congressional leaders and top administration officials, Obama said he was "modestly optimistic" the Senate leaders would reach an agreement. At the same time, he conceded, "Nobody's going to get 100% of what they want."
However, conservative activist Grover Norquist has vowed to back primary challenges against Republicans who violate his widely signed pledge not to raise taxes. Even if a deal is reached, Norquist has predicted yet more budget showdowns every time the government needs additional money to operate.
The House will reconvene Sunday, and the chamber's Republicans will get together sometime early Sunday night, according to a note sent Saturday to legislators and staffers.
Republicans have opposed any increase in tax rates, and Boehner suffered a political setback by offering a compromise -- a $1 million threshold for the higher rates to kick in -- that his GOP House colleagues refused to support.
The saga has fueled disdain for politicians by many Americans. Such contempt is deserved, said Rep. Steven LaTourette, an Ohio Republican, who is retiring from Congress.
"I think America should be embarrassed by its leadership in D.C.," he told CNN on Friday. "The fact that we have been unable to do things, and instead worried about our next elections. ... I think it's sinful."
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