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“My dad used to have a saying that, ‘Things that are alive swim against the waves. Things that are dead sort of float with the waves,’” said Ken Blackwell, former secretary of state in Ohio and fellow at the Family Research Council, adding that people tend to think that Mr. Cruz and Mr. Paul are more likely than Mr. Ryan to challenge the status quo in a conservative-movement fashion.

“If he is positioning himself to be the next speaker of the House when that becomes a possibility, that would explain some of his moves because I don’t think Paul or Cruz have any ambition to be the next majority or minority leader in the Senate,” Mr. Blackwell said. “That would set in my mind the basis for them to have two different approaches.”

Polls show voters put Mr. Ryan right in the middle of the scrum for the 2016 Republican presidential nomination.

“At this point, no one stands out from the pack and Ryan is part of the pack,” said Peter A. Brown, assistant director of the Quinnipiac University Polling Institute.

Mr. Ryan has a chance to bolster his stock Friday among Christian conservatives when he addresses the annual Values Voters Summit via video.

“Our job is to preserve our values in the 21st century,” Mr. Ryan will say, according to his office. “We need to completely rethink government’s role in helping the most vulnerable. We need to completely rethink government’s role in health care. That means we can never give up on repealing and replacing Obamacare.”

Mr. Cruz, Mr. Paul and Mr. Rubio, as well as former Sen. Rick Santorum, are scheduled to speak in person at the summit. Only Mr. Ryan, citing the budget talks and the government shutdown, has said he will appear via video.

Mr. Ryan also is scheduled next month to headline Iowa Gov. Terry Branstad’s annual birthday bash, putting him back in the first-in-the-nation caucus state.

Paul Ryan is a thoughtful leader and is the one person in Washington who has come up with a plan to get America back on track financially,” Mr. Branstad said. “I asked him to come to Iowa because I believe he has displayed excellent leadership in Washington in a time when tough decisions need to be made.”