House Speaker John Boehner declared that his rank and file generally stood ready to compromise in order to reach an agreement as a way of "getting our economy going again and growing jobs.
Still, 12 days before the default deadline, officials stressed that no compromise appeared imminent. And new hope of one ran instantly into old resistance: from Republicans opposed to higher taxes and Democrats loath to cut Medicare and other benefit programs.
While talks on a major, long-term agreement continued, a fresh, shorter-term backup plan appeared to be gaining momentum. Under discussion among some House Republicans, that proposal would cut spending by $1 trillion or slightly more immediately and raise the debt limit by a similar amount - enough to postpone a final reckoning until early in 2012.
Both sides maneuvered for political advantage and for leverage in negotiations about which little was publicly known.
"At the end of the day, we have a responsibility to act," said Boehner of GOP lawmakers,
Across the Capitol, however, Senate Majority Leader Harry Reid blamed some of the same Republicans - "tea party extremists," he called them - of blocking a deal.
The sometimes-conflicting information underscored the frenzied final days before a threatened default, when the Treasury would no longer be able to pay all its bills in full and the economy could go into a tailspin as interest rates spiked.
Some Democrats confided they were worried Obama would sign off on an agreement that cuts benefit programs without raising tax revenue, and they peppered Budget Director Jack Lew - in a closed-door meeting in the Senate - with questions about the high-level negotiations.
The president has said publicly he wants a deficit-reduction plan that includes both, and Reid pointedly reminded him that was the case.
"My caucus agrees with that - and hope the president sticks with that, and I'm confident he will," the Nevada Democrat said.
Within hours, he and other senior Democrats in both houses were on their way to the White House to meet again with Obama.
One official said the White House had notified Democratic congressional leaders Wednesday night that Obama and the House leaders appeared to be were closing in on a deal that is said to include $3 trillion in spending cuts but only a promise of higher revenues.
Boehner's office and the White House said that account as overblown in part and inaccurate in part. Some Republicans charged it had been spread to anger Democrats and torpedo any possibility of a deal that would cut Medicare or Social Security.
"We're not close to a deal," said Jay Carney, the White House press secretary.
Another Democratic official said that in fact progress had been made, but Boehner's office declined to say as much.
"While we are keeping the lines of communication open, there is no `deal' and no progress to report," said spokesman Michael Steel.
The government's debt stands at a record $14.3 trillion and has been growing by more than $1 trillion a year. Obama's request for an increase prompted Boehner to say months ago that any rise must be accompanied by spending cuts of at least the same amount.
Publicly, some Republicans insisted they would not entertain any fallback measure as long as a separate House-passed bill was pending in the Senate.
That measure would raise the debt limit by $2.4 trillion while requiring an estimated $6 trillion in cuts and a congressional vote to send the states a constitutional balanced budget amendment for ratification.
Sen. Jim DeMint, R-S.C. an influential conservative, said that bill was "the only one that can pass before the Aug. 2 deadline."
But Reid said the legislation "doesn't have one chance in a million of passing the Senate," and privately senior Republicans in both houses were operating on the assumption that it would fail when the vote was taken.
Initially, that vote was set for Saturday, to be followed by an unveiling of an earlier fallback plan crafted by Reid and Senate Republican leader Mitch McConnell of Kentucky. At mid-day, though, Reid announced without explanation the Senate would formally reject the House-passed bill on Friday.
In another measure of the unpredictable nature of the debate, still another proposal - a bipartisan deficit-reduction plan that drew surprising support from Republican senators earlier in the week - appeared to be running into obstacles.
That plan includes $1 trillion in what its authors delicately call "additional revenue" in order to achieve overall deficit cuts of nearly $4 trillion.
Some Republicans say that means tax increases, a deal killer as far as they are concerned.
Obama latched onto that "Gang of Six" blueprint earlier in the week to announce a final stab at a "grand bargain" agreement.
In fact, though, he and Boehner have been in negotiations for weeks, despite setbacks, and in recent days, House Majority Leader Eric Cantor has joined the talks.
The two GOP leaders met with the president at the White House on Wednesday night. Afterward, the White House contacted Reid and House Democratic leader Nancy Pelosi with an update, according to numerous officials.