Although the latest GOP counter-offer falls far short of what Barack Obama is looking for, it suggests the two sides are edging towards a compromise.
The Republicans reported that Obama, too, is showing signs of a willingness to bend since his initial proposal, which included $1.6tn in new revenue. A new White House offer delivered on Monday cut this to $1.4tn, according to GOP figures, though this is still way above the $800m they are ready to accept.
Many members of Congress are hoping that a deal will be in place by Friday next week so they can head off for a long, uninterrupted Christmas and New Year break.
But Obama and the Republican leadership would have to reach agreement by early next week to allow time for the legislation to be passed by both the Senate and the House by next Friday.
The Democratic leader in the Senate, Harry Reid, said the timeframe was feasible but would be tight.
"I think it's going to be extremely difficult to get it done before Christmas, but it could be done," he told reporters.
Speaking before details of the Republican counter-offer emerged, Reid said: "We can do things very quickly, but this is not something we can do easily, at least as far as bill-drafting goes. But until we hear something from the Republicans there's nothing to draft."
Shortly afterwards, the office of the Republican House Speaker, John Boehner, announced the counter-offer, though without disclosing the details.
"We sent the White House a counter-offer that would achieve tax and entitlement reform to solve our looming debt crisis and create more American jobs," a spokesman for Boehner, Michael Steel, said. "We're still waiting for the White House to identify what spending cuts the president is willing to make as part of the balanced approach he promised the American people."
Obama and Boehner met at the White House on Sunday to work on a deal.
The Republicans, having been strenuously opposed to any tax rises, appear to be ready to accept an increase in taxes for the wealthy proposed by Obama. But they are expressing unhappiness that Obama is not prepared to match this with deep cuts in federal spending, particularly on entitlement programmes.
In a rare intervention from the House floor at the start of business on Tuesday, Boehner called on Obama to identify specific spending cuts. "Where are the president's spending cuts? The longer the White House slow-walks this process, the closer our economy gets to the fiscal cliff," Boehner said.
He added: "Well, here's what we do know. We know that the president wants more 'stimulus' spending and an increase in the debt limit without any cuts or reforms. That's not fixing our problem. Frankly, it's making it worse.
"And on top of that, the president wants to raise tax rates on many small business owners. Now, even if we did exactly what the president wants, we would see red ink for as far as the eye can see. That's not fixing our problem, either – it's making it worse and it's hurting our economy."
If there is no deal in place by January 1, sharp tax rises and cuts in spending will automatically kick in.
The White House spokesman Jay Carney expressed hope that a compromise will emerge. "There is a deal out there that is possible," he told the daily White House briefing.
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