The overarching problem for the man at the center of the budget fight, say allies and opponents, is that he and his leadership team have no real idea how to resolve the fiscal showdown.

They are only trying to survive another day, Republican strategists say, hoping to maintain unity as long as possible so that when the Republican position collapses, they can capitulate on two issues at once — financing the government and raising the debt ceiling — and head off any internal party backlash. Republican lawmakers say Mr. Boehner has assured them privately that he will not permit a default.

Backers of the speaker say he does not have to fear a coup. His obvious successor, the majority leader, Representative Eric Cantor of Virginia, is so caught up in the current legislative battle that he would probably be washed away with Mr. Boehner in a Tea Party putsch, and with no other obvious candidates in waiting, no such uprising is likely.

“I do not believe he’s primarily concerned with saving his speakership. He’s concerned with saving the House Republican Conference,” said Vin Weber, a former Republican House member who advises the current House leadership…

Asked what the House was doing, Representative Devin Nunes, Republican of California and a Boehner loyalist, said: “You really have to call Cruz, I’m not even joking about that. That’s really what you have to do, because he’s the one that set up the strategy, he’s the one that got us into this mess, and so we’ve got to know what the next move is.”

Democrats say they simply cannot trust the speaker to deliver.