Few veteran courtiers of the nation’s most prominent modern Republican dynasty left the gathering doubting that Mr. Bush was giving it serious consideration. Where a few months ago many close to the Bush family thought the second son was not planning to run, now many of the same people said he seemed more engaged in the possibility and making moves they interpreted as laying the groundwork.

“He sounds like a candidate to me,” said Joe Hagin, who worked in the White House for both President Bushes. “People are standing back and saying the country is ready for somebody who’s serious.”

Andy Card, who likewise worked in both administrations, including as George W. Bush’s chief of staff, said Republicans should draft Jeb. “If Jeb Bush doesn’t run for president, shame on us,” he said. “He is demonstrating the kind of leadership we desperately need both in our party and in our nation.”