Leaders of both parties continued to negotiate over the telephone, hoping to settle the matter before the opening of the Asian markets later on Sunday. That opening is widely feared to be the first real test of the tangible financial market fallout from the debt limit impasse. It was far from clear that such a deal would or could be reached in time.

As the Aug. 2 deadline for lifting the debt ceiling nears, warnings are growing that the nation’s economy may be damaged because of the protracted political stalemate. A downgrade of the nation’s credit rating, which could raise the cost of borrowing, seemed more likely, deal or no deal.

The impasse has further set the tone for the 2012 presidential race, with the debate growing rancorous and the two parties’ visions for the country at odds. The enormous deficit challenges have been clearly articulated to voters, as the parties seek unified control of Congress, where the Republicans are the majority in the House and the Democrats in the Senate.

“My last offer is still out there. I’ve never taken my last offer off the table,” Mr. Boehner said on “Fox News Sunday.” He said that plan included about $800 billion in new tax revenue as well as significant spending cuts.

“The preferable path would be a bipartisan plan that involves all the leaders, but it is too early to decide whether that’s possible,” Mr. Boehner added. “If that’s not possible, I and my Republican colleagues in the House are prepared to move on our own.”

The contours of such a plan were far from clear at midday Sunday. But it seemed likely to take the form of a two-step process, with a short-term increase in the debt limit along with about $1 trillion in cuts, an amount the Republicans said was sufficient to clear the way for a debt limit increase through year’s end. That would be followed by future cuts guided by a new legislative commission that would consider a broader range of trims, program overhauls and revenue increases.

Both sides would also be given another chance to vote on a balanced budget amendment that would set stringent guidelines for future debt ceiling increases.

While the White House remains adamantly opposed to a two-step deal that does not extend the debt ceiling beyond next year’s elections, administration officials expect that the Senate would modify Mr. Boehner’s proposal.

At the moment, the White House is a spectator to the 11th-hour game plan, after Mr. Boehner’s abrupt abandonment of negotiations on Friday night.

While a modified plan might fail to gain the support of the more right-leaning and Tea Party-influenced House members, it could win enough Democratic votes to pass if it is blessed by the Senate majority leader, Harry Reid. However, if Mr. Boehner were to reject Senate modifications and go with a deal that would pass muster with his Republican conference, the Senate would have to offer a rebuttal, as the clock ticks.

Such a plan has been rejected out of hand by the White House and Senate Democrats as unacceptable; they prefer one that would clear the way for roughly $2.5 trillion in extra borrowing needed to get through 2012, just beyond the presidential election.

Indeed, the so-called grand bargain between Mr. Boehner and Mr. Obama has not been left for dead — as have plans crafted by Mr. Boehner; the Republican minority leader, Mitch McConnell; and others — because, at the end of the day, the $800 billion in tax revenues and trillions of dollars in cuts amount to perhaps the best deal either party can get, even if it is unacceptable to the base of each.

Mr. Obama has said he will veto any debt legislation unless it extends the ability of the nation to borrow into 2013, the White House chief of staff, William M. Daley, said on Sunday.

Mr. Daley, appearing on the NBC News program “Meet the Press,” said that world markets and the American economy could not continue with the doubts brought on by periodic fights over the debt ceiling. “The president believes that we must get this uncertainty out of the system,” Mr. Daley said.

Minutes after Mr. Daley promised a veto, Senator Tom Coburn, Republican of Oklahoma, said on the same program that Mr. Daley’s remarks were “a ridiculous position, because that’s what he’s going to get presented with.”

Meanwhile, Treasury Secretary Timothy F. Geithner suggested there was still hope for a grand bargain. “They are talking again,” Mr. Geithner said in an appearance on “This Week” on ABC. “They’ve been in touch throughout this time.”

Both Mr. Geithner and Mr. Daley said they believed Congress would figure out a way to avoid default. “The leaders of Congress have said unequivocally that we will meet our obligations,” Mr. Geithner said.

Mr. Daley acknowledged, however, that both sides were still at the brink. “We are now getting to a point where markets around the world will question whether the political system can come together and compromise for the good of the country,” Mr. Daley said.

On Saturday, Congressional leaders clashed as they tried to reach a new deficit reduction agreement that Mr. Boehner told colleagues could cut $3 trillion to $4 trillion in spending over 10 years.

“We are working, and I’m confident there will be resolution,” Mr. Boehner told his fellow House Republicans in an afternoon conference call, according to participants. “There has to be.”

Maria Newman contributed reporting from New York.

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