With concerns growing that global financial markets could be thrown into turmoil if Congress does not agree to raise the debt ceiling, Republicans said they did not know whether Mr. Boehner would have enough support from the most conservative members in his conference to put a Senate plan up for a vote — if the leaders reach a deal.

“The question is: What will Senate Republicans do, what will Senate Democrats do?” said Representative Adam Kinzinger, Republican of Illinois.

Many Republicans said that however frustrated they were that the White House would not negotiate with them, they were just as dismayed with House colleagues who would not back down from their demands that any deal include provisions to chip away at the health care law.

“The problem here is that we don’t have a functioning majority,” said Representative Devin Nunes, Republican of California. “After three weeks of this, they’re still not figuring it out. I don’t know what it takes.”