"We expect further conversations tonight," Majority Leader Eric Cantor, R-Va., said cryptically, after he, Speaker John Boehner and a delegation of House Republicans had met for an hour or so with Obama at the White House.
He spoke at dusk, long after Boehner first outlined a compromise proposal that the White House carefully avoided rejecting - only to have Senate Democrats declare it unacceptable.
"Not going to happen," Majority Leader Harry Reid said brusquely. The Republican plan would leave the shutdown in place while raising the nation's $16.7 trillion debt limit and setting up negotiations between the GOP and Obama over spending cuts and other issues.
Heartened by any hint of progress, Wall Street chose to accentuate the positive. After days of decline, the Dow Jones industrial average soared 323 points on hopes that the divided government was taking steps to avoid a default. Reid's dismissive comments at the White House came at the end of the trading day.
Senate Republicans forged ahead on an alternative of their own that would ease both the debt-limit and shutdown crises at once. Officials said that it would require Obama to agree to some relatively modest changes to the health care law that stands as his signature domestic achievement.
The up-and-down day coincided with a dour warning from Treasury Secretary Jack Lew, who told lawmakers that the prospect of default had already caused interest rates to rise - and that worse lay ahead.
Appearing before the Senate Finance Committee, Lew said the Treasury must pay Social Security and veterans benefits as well as salaries to active duty military troops during the second half of this month. He said failure to raise the debt limit by Oct. 17 "could put timely payment of all of these at risk."
House Speaker John Boehner led a delegation of fellow Republicans to the White House for a late-afternoon meeting with Obama as the two sides groped for a way out of the latest in a string of crises. House Majority Leader Eric Cantor later pronounced the meeting useful and said there would be further conversations.
"I would hope the president would look at this as an opportunity and a good faith effort on our part to move halfway, halfway to what he's demanded, in order to have these conversations begin," Boehner, R-Ohio, told reporters earlier in the day.
He spoke after informing his rank and file that he intends to bring legislation to the floor as early as Friday to let the Treasury borrow freely until Nov. 22, contingent on Obama's agreement to open talks on legislation to reopen the government and discuss other pressing issues.
That would leave in effect the partial government shutdown, in its 10th day Thursday, that has idled 350,000 federal workers but so far has not produced the type of widespread economic hardship that a default might mean.
After more than a week of lost tourism, some governors prevailed on the Obama administration to let states use their own money to pay for national parks to reopen, Grand Canyon and Zion among them. There was a catch - the Interior Department made it clear it didn't plan to reimburse the states after the shutdown ends.
Some tea party-aligned lawmakers claimed partial credit for the GOP retreat, casting it as a way of finessing one problem so they could quickly resume their own campaign to deny operating funds for the national health care overhaul known as "Obamacare."
At the White House, spokesman Jay Carney told reporters the president would "likely sign" a short-term extension in the debt ceiling, and did not rule out his doing so even if it left the shutdown intact.
Ironically, Boehner's plan stirred grumbling among relatively moderate Republicans who said the shutdown should end, but little if any unhappiness among the staunch conservatives who often part company with party leaders.
One Republican said he and fellow tea party allies deserved at least partial recognition for the plan that would raise the debt limit without reopening the government.
"I actually went to (Majority Leader) Eric Cantor a couple days ago and I proposed this. I said, 'You're going to think this is crazy but I, as a conservative, would be willing to vote for a debt ceiling for six weeks.," said Rep. Raul Labrador, R-Idaho.
Additionally, Sen. Ted Cruz, the conservative Texan who has played a prominent role in this fall's budget struggles, raised no objections.
Interviewed on radio KFYO in Lubbock, Texas, he said, "My understanding is this is being driven by House conservatives who are quite reasonably saying, 'Listen, let's focus on Obamacare, on winning the fight on Obamacare ... and let's push the debt ceiling a little further down the road so that it doesn't distract us from the fight we are in the middle of right now."
For his part, Reid has proposed no-strings-attached legislation to raise the debt limit by $1.1 trillion, enough to prevent a recurrence of the current standoff until after the 2014 elections.
In remarks on the Senate floor during the day, Sen. Mitch McConnell of Kentucky, the GOP leader, said that Democratic measure "just won't fly. ... The American people can be persuaded to raise the debt ceiling, but they're not in any mood to simply hand over a blank check."
Since the current standoff began more than two weeks ago, Republican demands have shifted continuously, while the president's position has remained essentially unchanged.
The shutdown began on Oct. 1 after Obama ruled out any concessions that would defund, delay or otherwise change the new health care law. He said he would be willing to negotiate on a range of issues, but only after the shutdown was ended and the debt limit raised.
For their part, Republicans drafted a long list of demands to accompany any increase in the debt limit, including some that would raise the cost of Medicare for better-off beneficiaries, make changes to the health care law and roll back several environmental regulations either issued or in the planning stages by the administration.
In recent days, the focus has shifted from the shutdown to the threat of default, and Republicans have spoken less and less frequently about insisting on concessions in the health care law.
The call for negotiations on long-term deficit cuts would mark a return to basics for the House Republican majority.
Shortly after taking control in 2011, the rank and file initiated a series of demands to cut spending, culminating in an agreement with Obama that cut more than $2 trillion over a decade.
After four years of trillion-dollar deficits, the 2013 shortfall is expected to register below $700 billion.
At the same time, the nation's debt rises inexorably. It was $10.6 trillion when Obama took office during the worst recession in decades, and has grown by $6.1 trillion in the years since.
Associated Press writers Alan Fram, Stephen Ohlemacher, Andrew Taylor, Jim Kuhnhenn, Julie Pace, Donna Cassata and Martin Crutsinger contributed to this story.