People at the fundraiser recalled that the former president said that Islamic women should come to the U.S. to experience a free society so they can lead the charge for equality in the Middle East. “He said everyone wants to be free,” said U.S. Rep. Brad Wenstrup of Cincinnati. “He really touched everyone in the room.”

Asked about the future of the Republican Party, Mr. Bush said, “As long as everyone feels welcome, I think we’ll succeed,” according to Mr. Wenstrup.

Mr. Bush also stressed to Mr. Portman’s donors that the institution of the presidency was more important than the occupant of the White House. Criticizing President Obama, he said, would demean the institution. Even in troubled times, “we’re lucky because we’ll always have the presidency,” Mr. Bush said, according to one attendee.