Washington (CNN) -- They went around and around for an hour, getting nowhere.
Then, suddenly, a change that could help break the logjam that's resulted in a partial government shutdown and the looming threat of a further crisis when the nation bumps up against its self-imposed borrowing limit.
Rep. Paul Ryan, the Republican chairman of the House Budget Committee, "said something to the effect of, 'Look, we know you don't like our position, we know you probably don't respect our position, but we're the Republican majority," CNN correspondent John King reported Friday on CNN's "New Day."
"'You're stuck with us for a while, at least through the next election season, so we need to learn to have a conversation with each other.'"
"And at that point, both Democrats and Republicans say, the tone of the meeting changed," King said. "The president said, "'Listen, I'm not going to negotiate with you until you reopen the government, but go to your members, find out what you need to do to get that part done and let's try to make some progress.'"
CNN's reporting on the meeting is based on accounts from multiple sources who attended.
The meeting appeared to help begin to break the logjam that has kept parts of the government shut down for 11 days, leaving hundreds of thousands of federal employees without work and causing countless ripple effects, from lost tourist revenue around national parks to a threat to the Alaskan crab fishing season.
"We're all working together now," Rep. Pete Sessions, a Texas Republican, said after the meeting.
House Majority Leader Eric Cantor called the meeting "very useful."
And the Obama administration described the session as "good" -- saying the President listened to Republican proposals and the two sides discussed "potential paths forward." Obama is supposed to meet with Senate Republicans Friday morning.
Beginnings of a deal?
After initially demanding changes to or the elimination of Obama's signature health care reform plan, Republicans have more recently focused on extending the debt ceiling for up to six weeks while negotiating on spending and other issues during a continued government shutdown.
Democrats have insisted that the debt ceiling be raised and the government reopened before they would be willing to negotiate on other issues.
Obama made clear, during his session with House Republicans, that he won't give concessions to reopen the government, according to a Democratic source familiar with the meeting.
But he has indicated willingness to consider a short-term debt deal, a Democratic lawmaker told CNN.
"If that's what (House Speaker John) Boehner needs to climb out of the tree that he's stuck in, then that's something we should look at," according to the lawmaker, who attended the meeting and spoke on the condition of anonymity.
"There was not a timeline set," Rogers said. "But we want to move quickly."
Some congressional Democrats have balked at the outline of the GOP offer, insisting the government must reopen and the debt ceiling must be increased to get broader talks going.
"One way or another both of those have to happen," said veteran Sen. Carl Levin of Michigan.
Another Democrat, Rep. Elijah Cummings of Maryland, said Obama "needs to press for the opening of the government."
"Without a doubt the default would be much more catastrophic, but I've got constituents, a lot of whom work for the federal government who are going through catastrophes every hour," Cummings said.
Anti-Obamacare provisions no longer in GOP plans
One thing any agreement won't include, it appears, is a provision targeting the 2010 Affordable Care Act.
Ryan, who was the Republican party's vice presidential nominee last year, didn't mention Obamacare in a Wall Street Journal op-ed, saying instead that politicians from both parties should focus on "modest reforms to entitlement programs and the tax code."
As others have done in recent days, GOP Rep. James Lankford of Oklahoma told CNN on Thursday that going after the President's signature health reform is "currently off the table."
Yet Sen. Ted Cruz of Texas -- one of the most vocal Republicans in the debate -- wasn't so ready to let the health care issue go. Speaking Thursday on CNN's "Crossfire," he said, "Democrats in this town do not want to discuss Obamacare."
On Obama's signature health care reform and what's happening in Washington generally, Cruz said that it's House Republicans who "are listening to the millions of Americans" to do what's best for them.
According to a GOP source, it's not certain whether Boehner can gain support from some or most of his GOP caucus for a plan without anything to do with Obamacare or other concessions. That could mean, if a proposal like the one floated Thursday proceeds, the Speaker may need Democratic votes to pass it.
Failure to raise the debt ceiling by next week's deadline would leave the government unable to borrow money to pay its bills for the first time in its history. And absent a breakthrough, the shutdown would continue at a cost estimated at up to $50 billion a month.
All of this is taking a toll on Washington's reputation: A national CNN/ORC International survey released Monday indicated that Americans are blaming all parties in the fight, though Republicans got the worst of it.