Their concerns have prompted the late emergence of two experienced candidates, Deval Patrick, the former governor of Massachusetts and a close friend of Mr. Obama, and Michael R. Bloomberg, both of whom entered the field after nine candidates had already dropped out.

Those worries are not necessarily shared by voters in early states, where surveys show that most Democrats have gravitated to a short list of four candidates and feel the field remains too large.

“If you talk to the voters they don’t feel the need for another candidate,” said Kathy Sullivan, the former chairwoman of the Democratic Party in New Hampshire. “We haven’t been sitting around in New Hampshire saying we need a hero to come and rescue us.”

Since leaving office nearly three years ago, the former president has mostly remained out of the public fray.

While he’s headlined fund-raising events for Democratic committees, the former president has kept a low profile in the presidential race, meeting privately with most of the Democratic candidates but telling friends and political allies that neither he nor his wife, Michelle, have plans to endorse anyone and do not see it as their place to steer the party’s future.

Mr. Obama has expressed interest, at different times, in rising stars like former Representative Beto O’Rourke of Texas and Mr. Buttigieg. Mr. Obama had issued a warm statement about Mr. Biden’s entry into the primary but had declined to endorse him or anyone else.

During his nearly hourlong remarks on Friday, Mr. Obama offered only one clear endorsement, though not exactly the one many Democratic donors in the room wanted to hear.