“This is a cycle in which a younger generation of politicians are coming into the race with a view that small-government conservatism is the ideal and who feel no imperative to bend over backward to show that they are compassionate to people,” said Ben Domenech, a conservative who writes a daily newsletter popular on the right.

Mr. Bush and Mr. Huckabee have each taken positions that are intended to broaden the party’s appeal but that will inflame the Republicans’ more doctrinaire conservatives.

Both backed the Common Core education standards, now detested among many activists because they are seen as opening the door to more federal control. Both have taken a comparatively temperate approach on immigration. Mr. Bush has opposed offshore drilling, and Mr. Huckabee, in the 2008 primary, stated his support for a cap-and-trade program to reduce carbon emissions. Mr. Huckabee has also battled for years with the fiscally conservative Club for Growth because he increased spending and some taxes when he governed Arkansas.

“If Bush and Huckabee run their campaigns as if they are just picking up from when they left office at the end of 2006, they won’t succeed,” said William Kristol, editor of The Weekly Standard. “But they are both smart enough to probably understand that.”

Each has taken steps to accommodate today’s Republican Party. Mr. Bush has not retreated from his support for the Common Core, but after mocking those in his party who reversed their position on the standards in April he recalibrated his view in a November speech, saying, “Nobody in this debate has a bad motive.”

And in unveiling his political action committee this week — promoted via “Smartphone Video,” as his website noted — Mr. Bush posted a message that mixed his commitment to an immigration overhaul and education standards with pledges to “protect liberty” and “re-limit government.”

“We believe the income gap is real, but that only conservative principles can solve it by removing the barriers to upward mobility,” he wrote.