Boehner will brief his colleagues at an all-important House conference meeting tomorrow morning. In deciding whether to spurn McConnell, a key issue for Boehner is whether he can get his conference to back something that’s sufficiently modest it’s seen as a plausible compromise by Democrats and the public.

“That is the challenge, is to figure out how to thread the needle between getting any 218 votes here that we can also get whatever it is closed to passing or being seen as reasonable on the other side of the Capitol,” Stivers says.

Timing may play an unexpectedly critical role. If the Senate begins proceeding on a debt-ceiling bill tomorrow, it could be as long as Saturday before it passes that chamber, given its procedural rules. In that time, during which the debt-ceiling deadline will come and go, Boehner could go on offense.