“I think all the other candidates will take him seriously,” Mr. Nutter said.

Mr. Biden, who appeared to have the most to lose from the sudden entry of a prominent centrist with unlimited financial resources, spoke guardedly to reporters in New Hampshire about that possibility on Friday. He called Mr. Bloomberg an “honorable guy” and deflected a reporter’s question about whether Mr. Bloomberg, who was elected mayor of New York City as a Republican and became a Democrat only last year, could be considered an authentic member of the party.

“Michael’s a solid guy, and let’s see where it goes. I have no, no problem with him getting in the race,” Mr. Biden said in New Hampshire, adding, “Last polls I looked at, I’m pretty far ahead.”

Advisers to Mr. Biden have argued in private that Mr. Bloomberg should not be overestimated as a threat to the former vice president, because he may struggle to appeal to the African-American voters and working-class whites who make up most of Mr. Biden’s political base.

Mr. Bloomberg, who is one of the country’s richest men, has his own vulnerabilities, such as his support for stop-and-frisk policing and charter schools, and his opposition to liberal economic policies that involve taxing large private fortunes and breaking up banks and other big corporations.

President Trump, for his part, did not sound concerned about Mr. Bloomberg on Friday morning. Mr. Trump noted that Mr. Bloomberg would spend a lot of money on a campaign and predicted he would damage Mr. Biden in the process. Speaking in the terms of derision he usually applies to political rivals, Mr. Trump said without elaborating that Mr. Bloomberg had “personal problems” and applied a derisive nickname.

“There’s nobody I’d rather run against than little Michael,” Mr. Trump said.

Ms. Warren and Mr. Sanders have already begun to take on Mr. Bloomberg more assertively, both emailing their supporters to warn that a prominent billionaire was seeking to — as they put it — buy the presidency. “The billionaire class is scared and they should be scared,” Mr. Sanders tweeted on Thursday evening, without mentioning Mr. Bloomberg by name.

Campaigning in Raleigh, N.C., on Friday, Ms. Warren avoided attacking Mr. Bloomberg directly but told reporters that a candidacy funded from personal wealth would not produce “a government that works for the people.”