Hillary Clinton could have shared the Democratic presidential ticket with Bill Gates — or with his wife, Melinda. She could have named Michael R. Bloomberg, the billionaire former New York mayor, as her running mate. Or she could have chosen to reward a loyal friend and fund-raiser by making Gov. Terry McAuliffe of Virginia her No. 2.

All four were among the potential vice-presidential candidates Mrs. Clinton’s top campaign aides included in a first pass through their screening process, which turns out to have been well underway even before she amassed the delegates she needed to capture the nomination.

On March 17, just after Mrs. Clinton had defeated Senator Bernie Sanders of Vermont in a string of primary contests, including those in Florida and Illinois, her campaign chairman, John D. Podesta, sent her a note of congratulations.

“Ok, I can breathe again!” he wrote in a hacked email released by WikiLeaks on Tuesday. “Congrats on a fabulous night. I am feeling like it’s possible to get back to the longer term again.”