Mr. Boehner, who plans to leave Congress by Oct. 30, has little time to act if he wants to ensure the debt ceiling doesn’t land on Mr. Ryan in his first week on the job. House Republicans are expected to select Mr. Ryan as their nominee for speaker on Wednesday, with his election on the House floor scheduled for Thursday.

“If the conference cares about Paul Ryan’s ability to succeed, we do need to do that,” Rep. Frank Lucas (R., Okla.) said of resolving the debt-limit standoff.

House GOP leaders have struggled to find the roughly 30 Republican votes needed to pass a debt-ceiling increase. They shelved a proposal this past week from the Republican Study Committee, a large group of House conservatives, that would have tied an increase to large spending cuts and a freeze on new federal regulations.

A House debt-limit increase tied to other conservative policies would probably not come back from the Senate, shorn of those measures, until after Mr. Boehner’s departure.