But advisers to Mr. Bloomberg, 77, say he is also concerned that Mr. Biden may not be strong enough to win the nomination, and he believes that Ms. Warren or Mr. Sanders are too liberal to defeat Mr. Trump.
The president, 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.
“I don’t think that big money ought to be able to buy our elections, and that’s true whether we’re talking about billionaires or corporate executives that fund PACs or big lobbyists,” Ms. Warren said.
In a sign of the pressure Mr. Bloomberg may face to begin campaigning quickly, the Democratic state chairmen in Iowa and New Hampshire, the first two states to hold nominating contests, issued a joint statement on Friday strongly suggesting that Mr. Bloomberg should plan to make himself available to voters there soon.
Ray Buckley, the New Hampshire chairman, and Troy Price, the Iowa chairman, noted that the primary field included “so many qualified candidates” with progressive ideas — a nod to the number of candidates who have already been working over the early states for many months.