The test will be whether Mr. Ryan — who declined last year to sit on another Congressional committee charged with taming the deficit, in large part because doing so might have hurt his prospects for national office — can make the transition from House budget philosopher to governing heavyweight who can help negotiate a bipartisan deal and sell it to his colleagues.

While President Obama and the Democrats are expected to give ground on entitlements and discretionary spending, it is likely that Mr. Ryan will be the player under the most pressure to back away from his previous conservative positions in order to form a bipartisan agreement.

Mr. Ryan, who declined to be interviewed, was largely silent during the campaign about his call for changes to the Medicare system and for vast cuts to government services, as outlined in his House budget. But his distaste for Mr. Obama’s fiscal theories was unambiguous. At the Republican convention, for example, he called the Obama administration’s economic vision “a dull, adventureless journey from one entitlement to the next, a government-planned life, a country where everything is free but us.”

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.”