With his new muscle and increased respect from his colleagues, Mr. Ryan could conceivably scuttle any deal if he loudly opposes a solution that the speaker and the top Republican leaders embrace. But his conservative base might rebel against him if he were to endorse any deal seen as awarding too much to Mr. Obama and the Democrats, particularly on tax rates. Some Republicans think the pitfalls are dangerous enough that Mr. Ryan might consider leaving Congress altogether to work on his policy agenda without the inherent headaches of the Hill.

“He has to think about what he wants his role to be,” said Representative Tom Cole of Oklahoma. “Is he going to run in 2016, or run for something else in Wisconsin, or play a bigger role in the House? He’s going to play an outsize role here because of the national profile he now has, but on the other hand, this conference is quite happy to act independently.”…

While Ms. Murray noted that Mr. Boehner was the official leader of the House Republicans and that she found the speaker’s recent conciliatory tone “encouraging,” she said Mr. Ryan’s impact was a wild card. “If he is bringing Ryan in to have his budget of the past as the present stance,” she said, “it is going to be long row to hoe. If they are bringing Ryan in to listen and compromise, that will represent a path forward.”