House Speaker John Boehner summoned rank-and-file conservative lawmakers in a Sunday-afternoon conference call to be ready for compromise that is "going to require some of you to make some sacrifices."
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He said the new legislation, expected to be unveiled as early as Monday morning, would cut federal spending, raise the government's borrowing authority and be able to clear both houses of Congress, according to one official familiar with his remarks on the call.
Officials said that plan was likely to provide for an immediate increase in the debt limit of $1 trillion, while making somewhat larger cuts in federal spending. Additional spending reductions and greater borrowing authority would depend on future action by Congress - a two-step process that Obama has ruled out.
Across the Capitol, Senate Democratic leader Harry Reid worked on an alternative to cut spending by $2.7 trillion and increase the debt limit by $2.4 trillion - enough to satisfy Obama's demand that the current crisis not recur before the 2012 elections.
Reid said the plan "meets Republicans' two main criteria" - spending cuts greater in size than the increase in borrowing authority and no new revenue.
The events unfolded in a crisis atmosphere - in a sweltering city swarming with summertime tourists - a little more than a week before an Aug. 2 deadline for action by Congress to raise the government's $14.3 trillion debt limit.
Without action by that date, the Treasury will be unable to pay all its bills, possibly triggering a default that could have severe consequences for the U.S. economy and the world's, too.
Officials - and many stockholders - were holding their breath to see how the uncertain situation would affect financial markets in the U.S. and around the world as they opened after the weekend of crisis negotiations. Early results from Asian markets showed no major swings.
Obama spoke with Boehner by phone during the day, then met with Reid and House Democratic leader Nancy Pelosi at the White House.
At the same time, several officials said leadership aides spent Sunday trying to produce a compromise that could quickly clear both houses.
One official, speaking on condition of anonymity, portrayed the Democrats' White House meeting as an unsuccessful attempt to persuade Obama to accept a two-step he has said he won't agree to.
There were numerous suggestions of progress - both Boehner and Reid now support plans without immediate increases in government revenue - but no announced compromise.
Similarly, the precise elements of the rival fallback plans were sketchy.
Boehner's revised measure would fall far short of a House-passed bill that was rejected in the Senate late last week.
It had called for raising the debt limit on condition of $6 trillion in spending cuts and congressional approval of a constitutional balanced budget amendment for state ratification.
Reid's office declined to provide any details of his alternative, except to say it would cut $2.7 trillion in spending over a decade without any tax increases.
Any sum that large would require either cuts to benefit programs that aides said Reid preferred to leave untouched, or savings from the Pentagon, possibly by assuming the end of the wars in Iraq and Afghanistan.
It appeared the White House was largely consigned to a spectator's role.
Asked what the administration's plan was to avoid default, Treasury Secretary Timothy Geithner said, "Our plan is to get Congress to raise the debt ceiling on time."
The state of play veered between bipartisanship and brinkmanship on an issue of immense economic consequences.
Despite hours of compromise talks in the Capitol, lawmakers' aides had so far been unable to agree on a two-step plan that would satisfy Obama's demand for a large enough increase in the debt limit to tide the Treasury over until after the 2012 elections.
White House Chief of Staff Bill Daley said anything short of that would be a gimmick and prompt the world to say: "These people just can't get their act together."
Interviewed on Fox, Boehner said, "I would prefer to have a bipartisan approach to solve this problem. If that is not possible, I and my Republican colleagues in the House are prepared to move on our own."
White House and congressional leaders talked past each other on the Sunday TV shows as negotiations unfolded in secrecy.
"There will be a two-stage process. It's just not physically possible to do all of this in one step," Boehner said. "I know the president is worried about his next election. But, my God, shouldn't he be worried about the country?"
With an eye on the financial markets, Geithner insisted anew the United States would not default.
"It's just unthinkable," Geithner said. "We never do that. It's not going to happen."
The debt deal-making has consumed Washington for weeks and has put on display a government that at times risks utter dysfunction. Even after talks about between Obama and Boehner broke down in spectacular fashion Friday, Geithner said the two men were still negotiating.
He also suggested the ambitious framework the two leaders had discussed, targeting a deficit reduction of $4 trillion, remained under consideration.
"I don't know. It may be pretty hard to put Humpty Dumpty back together again," Boehner said of that grand plan. "But my last offer is still out there. I have never taken my last offer off of the table and they never agreed to my last offer."
Under any scenario, Washington's leaders have run themselves almost out of time.
Daley said, in fact, the consequences are already taking hold. "I don't think there's any question there's been enormous damage done to our creditworthiness around the world," Daley said.
Boehner appeared on "Fox News Sunday." Geithner was on Fox, ABC's "This Week" and CNN's "State of the Union." Daley and Coburn spoke on NBC's "Meet the Press," and Daley also appeared on CBS' "Face the Nation."
Associated Press writers Alan Fram, Stephen Ohlemacher and Nedra Pickler contributed to this report.