Mr. Bush opposes abortion, and he is no less an opponent of higher taxes than his brother, President George W. Bush, was in his two terms. However, he has refused to sign the antitax pledge of the conservative activist Grover Norquist, who helped lead the rebellion against his father when the elder President Bush broke his own “no new taxes” promise during his first and only term.

He could also benefit from what some Republicans see as a modest vein of Bush nostalgia, marked by a video shown at the Republican National Convention about his father, who is 88 and contending with a form of Parkinson’s disease and declining health.

Any political future for Jeb Bush depends on whether that warmer tide will be enough to offset lingering bad feelings about the family brand after the presidency of George W. Bush, who continues to be criticized by many conservatives for presiding over bailouts and expanding the size of government.

Still, calls for Jeb Bush to enter the arena in a bigger way represent vindication of a sort. His family’s longstanding advocacy for a more broad-based and “compassionate” Republican Party was largely ignored and eventually repudiated by the populist, small-government conservatives who held sway over the party after Mr. Obama’s election.

George W. Bush’s break with the populist right began midway through his second term over his support for a pathway to citizenship for some illegal immigrants, which grass-roots activists labeled an amnesty plan. His push for immigration legislation failed.

This year, even before Election Day, Jeb Bush was warning of what he called his party’s “stupid” approach to illegal immigration. (Mr. Obama took 71 percent of the Hispanic vote, according to interviews with voters.)

“The day after the election, I started getting e-mails and texts from friends and others wanting Governor Bush to run and asking whether he would,” said Justin Sayfie, a Florida lobbyist who served as an adviser to Mr. Bush when he was governor.