"Of all the bizarre moments" involved in the debate, said Rep. Lloyd Doggett, D-Texas, "this may be the most bizarre: that we will pay people not to work." He called it "the new tea party sense of fiscal responsibility."
House Republicans said they want to ease the pain from the partial shutdown. Democrats said Congress should fully re-open the government and let employees work for the pay they're going to receive.
Senate Majority Leader Harry Reid, D-Nev., said Saturday the Democratic-controlled Senate will approve retroactive pay for furloughed workers, although he didn't specify when.
Meanwhile Saturday, the Pentagon ordered most of its roughly 400,000 furloughed civilian employees back to work.
The politics of the 5-day-old partial government shutdown have merged with partisan wrangling over the graver issue of raising the federal debt limit by Oct. 17. If that doesn't happen, the White House says, the government will be unable to pay all its bills, including interest on debt. Economists say a U.S. default would stun world markets and likely send this nation, and possibly others, into recession.
Boehner, R-Ohio, and Obama say they abhor the idea of a default. But they and their respective parties have not budged from positions that bar a solution.
Obama says he will not negotiate tax and spending issues if they are linked to a debt-ceiling hike. Boehner and his GOP allies say they will not raise the ceiling unless Democrats agree to deep spending cuts.
Many House Republicans also demand curbs to Obama's signature health care law as a condition of reopening the government. The president and his allies call the demand absurd.
In interviews, key lawmakers and aides said they don't know how the impasses might be resolved. But they laid out several possibilities, all of which face huge political impediments.
(Read more: Washington deadlocked as shutdown enters fifth day)
-- Boehner yields. The speaker could pass bills to reopen the government and raise the debt ceiling -- with few or any concessions by Democrats -- if he decided to anger many conservatives in his 232-person caucus and rely heavily on Democrats' votes.
That's what Boehner did to mitigate massive tax increases at the beginning of the year and to give aid to victims of Superstorm Sandy. Most House Republicans opposed both measures.
But if Boehner were to enact something as contentious as a debt-ceiling hike with a "minority of the majority," he would face a GOP insurrection that could cost him the speakership. Many Democrats say he should do that.
Rep. Chris Van Hollen, D-Md., said Boehner "is going to have to decide to stand up to a reckless faction of his party for the good of the country. That's just the way this ends."
-- Both sides yield a bit. Democrats conceivably could offer a few concessions that might help Boehner attract a slim majority of his House Republicans. For instance, they could agree to lift a tax on medical devices that helps fund the new health law or approve the Keystone pipeline to carry oil from Canada.
(Read more: Shutdown halts IRS seizures)
Any such decisions, however, would violate Obama's repeated vow not to negotiate on the debt ceiling and the government shutdown.