“I hope he doesn’t enter the realm of dreamland and thinks he actually can get the nomination,” said Mitchell Draizin, a donor backing Mr. Biden, who has met with Mr. Buttigieg several times. “He could end up being the big winner out of this whole thing even though he won’t be president.”

That’s part of what infuriates his rivals, who acknowledge Mr. Buttigieg’s political talent but also see him as benefiting from certain advantages.

Over the summer, a simple mention of Mr. Buttigieg’s name during a conversation in the Senate chamber between Senator Cory Booker of New Jersey and Ms. Klobuchar was enough to make Ms. Klobuchar extremely agitated, according to witnesses.

Ms. Klobuchar, during an interview last weekend in Iowa, declined to answer when asked if Mr. Buttigieg is qualified to be president. Instead she pointed to his electoral record, which she said shows no signs he could beat President Trump in a general election.