Mr. Pence has neither the élan nor the bravado of Mr. Trump’s other vice-presidential finalists, Gov. Chris Christie of New Jersey and Mr. Gingrich. But Mr. Trump, a natural showman, has described Mr. Pence to associates as someone who looks the part, straight out of “central casting” with his clean-cut, silver-haired appearance. And Mr. Pence has admiringly called Mr. Trump “the people’s choice.”

“It is fair to describe the relationship as fairly new and quickly flourishing,” said Kellyanne Conway, a senior adviser and pollster to both Mr. Trump and Mr. Pence.

Still, close associates of the candidates say they are not a natural pair. David Kensinger, a Republican strategist in Kansas who has advised Mr. Pence in the past, said the Indiana governor was a privately charming man with a strict filter on the public stage.

“He tends to be very formal in public, and he had a reputation when he was in Congress of being a bit stiff,” Mr. Kensinger said. “Pence is a man of 21st-century ideas and 19th-century manners.”

Only a few weeks ago, the partnership between Mr. Trump and Mr. Pence might have come as a surprise to both men, who had no personal relationship. They have still met with each other only a handful of times.

An early and uncomfortable introduction, in the fall of 2011, came as Mr. Pence was preparing to run for governor and visited Trump Tower to seek a financial contribution from Mr. Trump. At the time, Mr. Trump was fascinated by gossip surrounding the marriage of Gov. Mitch Daniels of Indiana, whose wife had divorced him to be with another man, and then remarried Mr. Daniels several years later.