Both are also quite young. Mr. Needham is 36 and Mr. Rubio is 46. And both believe that the Republican Party has not done enough to rethink its animating ideas and appeal to voters at a time when Mr. Trump remains woefully unpopular with younger Americans.

In an interview on Tuesday, Mr. Rubio said that since his presidential campaign ended in March 2016, he has spent a lot of time contemplating what Mr. Trump got right and what he misjudged. And while Mr. Rubio insisted he had no plans to run for president again — “It’s so far-off in the future, I don’t know where my mind will be” — he said he believed that the conservative movement’s tenets need to evolve beyond sterile economic arguments of low taxes and high economic growth.

“I think the challenge is how the policies that come from those principles, by necessity, have to look different in the 21st century than they did in 1980 or 1985,” Mr. Rubio said. “America is not an economy. It’s a country,” he added, channeling some of Mr. Trump’s populism. “And we have a bigger job than just to increase G.D.P.”

Mr. Needham said he agreed with that diagnosis.

“Any fair-minded observer of the last several years would say conservatives have work to do in order to assure our principles remain relevant,” Mr. Needham said in an interview. “There was truth in candidate Trump’s declaration that this is the Republican Party, not the Conservative Party. Our challenge as conservatives is to build a movement that inspires a majority coalition of Americans.”

But beyond their shared views on the party’s need to have a better 20-year plan, the two have taken very distinct approaches to leadership. Mr. Needham has been a leading practitioner of the uncompromising, scorched-earth style of political combat that was a trademark of Tea Party-inspired politicians and activists. He frequently clashed with the Republican leadership in Congress and challenged it to drive a harder bargain on issues like defunding the Affordable Care Act, which led to a two-week government shutdown in 2013 that most Republicans came to see as ill advised.