Some congressional Democrats have balked at the outline of the GOP offer, saying 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.
One thing any budget resolution won't include, it appears, is provisions 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 to support 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 it is taking a toll on Washington's reputation: A national CNN/ORC International survey released Monday indicated that Americans were blaming all parties in the fight, though Republicans got the worst of it.