Five on the list were women — among them, former Gov. Jan Brewer of Arizona, Gov. Mary Fallin of Oklahoma, Sen. Joni Ernst of Iowa and Gov. Nikki Haley of South Carolina. Before the list was drawn up, Trump also expressed interest in Gov. Susana Martinez of New Mexico, but after Martinez did not return repeated phone calls from Lewandowski, Trump said that he was done with her — and then bashed the governor on a campaign stop in Albuquerque in late May. (Haley’s overt lack of interest in the job made her an early scratch as well.)

The last woman on the list was former Secretary of State Condoleezza Rice — a somewhat surprising inclusion, given Trump’s current disdain for the Iraq war with which Bush’s closest foreign-policy adviser will forever be associated. People “loosely affiliated” with the campaign, as Rice’s chief of staff Georgia Godfrey put it, paid a call to Rice and were informed that she had no interest in the job.

By early June, Manafort and Lewandowski had requested that Culvahouse do a formal vetting of six individuals on the list: Sessions, Ernst, Chris Christie, Mike Pence, Newt Gingrich and Senator Bob Corker of Tennessee. Ernst met with the nominee in Trump Tower, but her interest was only slight, and Trump’s seemed not much greater. She dropped off the list with no fanfare.

Corker was another matter. Though the much-respected Tennessee politician and former businessman had signaled his early interest in the job to Trump advisers, he did not share Trump’s taste for alley fighting. At a meeting earlier this month in Trump Tower, Corker separately told Manafort, Eric Trump, Ivanka Trump and her husband, Jared Kushner, that he didn’t think he was the best fit for the job. Corker then said the same thing to Trump, adding that he considered himself better suited for a cabinet position — say, Treasury or State Department — than as Trump’s brawling mate. Trump squinted at the senator for several seconds before saying, “O.K.” They then flew to North Carolina for a rally, where Trump called a tongue-tied Corker to the stage — leaving many observers to believe that this was the senator’s audition, and that he flunked it.

According to a prominent Republican who was regularly briefed on the vice-president considerations, Corker and Pence were the favorites, along with Kasich, who would soon be hosting the G.O.P. convention in Cleveland. But Kasich effectively removed himself from the list by telling Trump in a phone conversation at the end of May that a joint ticket would be like two corporations with completely different philosophies and styles trying to merge.

Trump nonetheless hoped that Kasich would at least formally endorse him by the time the convention began on July 18. On July 4, the Kasich adviser John Weaver called Manafort at the governor’s request and flatly informed him that no such preconvention endorsement would occur. Spurned twofold, Donald Trump would lose a vice president but gain an enemy.

Meanwhile, Trump’s final choice for the job, Mike Pence, did not hail from a swing state or arrive with presidential-campaign experience, as would have been the case with Kasich. But he was a Republican, and a governor, and popular among conservative evangelicals. Most important, he knew how to say “yes” to Trump.