On the Republican campaign trail, a grand compromise between Mr. Obama and Mr. Boehner would instantly become the defining issue of the campaign and would force difficult decisions on those running for the party’s presidential nomination.

Each of them would have to choose sides — either endorse the deal, and risk alienating the Tea Party movement that has energized voters, or come out against it and put themselves in opposition to their party leadership in Washington (a choice which Newt Gingrich, the former speaker of the House, found to be troublesome earlier this year).

Representative Michele Bachmann of Minnesota has already given her answer. In a new television ad set to run in Iowa, Ms. Bachmann said that she “will not vote to raise the debt ceiling.” It’s almost impossible to imagine her reversing that position if a deal is cut to do just that.

For the other Republican candidates, though, it might be a tougher decision.