The goal is to persuade thousands of the party’s most reliable patrons to overcome their lingering objections to the candidate most of them never wanted, and to help defeat a Democrat most of them want even less.

In the coming weeks, Mr. Trump and campaign officials will attend a string of high-dollar fund-raisers organized with the Republican National Committee, hitting the summer haunts of the well-to-do — from East Hampton to the California wine country — in a last-ditch effort to tap into the party’s vast financial reserves.

On Monday in Detroit, Mr. Trump is scheduled to unveil a set of detailed economic policy prescriptions, a speech his supporters hope will help Mr. Trump reset his campaign and remind wavering Republican donors of the stark contrast that he offers to Hillary Clinton on issues like taxes and regulation.

“It’s somewhat of a natural evolution with the donor base,” said Gaylord T. Hughey Jr., an energy lawyer who formerly backed Jeb Bush and is now helping lead Mr. Trump’s fund-raising in Texas. “I think the traditional donor base was somewhat shocked by Trump’s nomination. They’re learning more about him, but they’re also focused on what the alternative is, and what the implications of the alternative are.”

It is a dizzying turnaround for everyone involved, several donors said in interviews. Aides and fund-raisers for Mr. Trump, a self-described billionaire who has spent months proclaiming his independence from the party’s traditional financial interests, now concede that they need mainline Republican donors to swing behind Mr. Trump so that he will have enough financial firepower to compete with Mrs. Clinton in the air and on the ground.