“You were very mean to me!” Mr. Trump said, according to people familiar with the call. Mr. Trump was referring to Mr. Bloomberg’s slashing speech that year at the Democratic National Convention, during which he called the Republican a “con” and called on voters to elect a “sane, competent person.”

Mr. Trump settled down almost immediately, then turned the conversation toward his latest predicament: quickly hiring people to fill out his government.

Mr. Bloomberg, according to the people briefed on the call, told him that when he was first elected mayor in 2001, he, too, had never served in government. What Mr. Trump should do, Mr. Bloomberg advised, was to “hire a lot of people smarter than you.”

“Mike,” Mr. Trump replied tersely, “there is no one smarter than me.” A startled Mr. Bloomberg paused before turning the conversation to a less fraught subject: golf.