The meeting, initially expected on Friday evening, took place on Saturday morning at Trump National Golf Club in Bedminster, N.J., according to four people briefed on the get-together who requested anonymity because they were not authorized to speak about it.

Mr. Pence’s wife, Karen, was at the meeting, but Paul Manafort, the Trump campaign’s chief strategist who has been handling much of the search for a vice-presidential nominee, was not, two of the people briefed said.

One of those sources described the meeting as pleasant.

Mr. Pence, a favorite of religious conservatives who twice considered running for president, has risen in Mr. Trump’s circles in recent days as a possible nominee. Mr. Trump’s aides have started vetting him, one of the people briefed on the matter said.