While some of Mr. Bloomberg’s longtime advisers have committed to a campaign, including the pollster Doug Schoen and the ad-maker Bill Knapp, he would have to build a campaign from zero in the early primary and caucus states, and he may have a hard time qualifying for either of the two remaining debates this year.

In a Democratic race, Mr. Bloomberg would face a battery of complicated questions about his political ideology and governing record. He has been a vigorous advocate for core liberal causes, like gun control and battling climate change. But as mayor Mr. Bloomberg also championed police searches that targeted black and Latino men, and in an interview last fall, he defended his administration’s stop-and-frisk policing strategy.

In the same interview, Mr. Bloomberg also expressed skepticism about the #MeToo movement in general, as well as the specific allegations of sexual misconduct against Charlie Rose, the disgraced anchor who for many years broadcast his show from the offices of Mr. Bloomberg’s media company.

Mayor Steve Benjamin of Columbia, S.C., said in an interview that he spoke with Mr. Bloomberg in late October, and the two discussed the unfolding presidential race. Mr. Benjamin, a Democrat who has not taken sides in the primary, predicted Mr. Bloomberg’s background as a businessman and mayor, and his stances on climate and guns, would make him a formidable contender if he runs.

“There’s a lot of work to be done in gearing up for the first four primaries and caucuses, and certainly for Super Tuesday, but I would caution anybody not to underestimate Mike Bloomberg,” Mr. Benjamin said.

Ms. Raimondo, the Rhode Island governor, stopped short of backing Mr. Bloomberg as a presidential candidate but issued a glowing statement about his record.

“While this is not an endorsement, Michael Bloomberg is a friend and I admire his track record as a successful business leader and mayor who finds practical solutions to some of America’s biggest challenges, from creating good jobs to addressing the opioid crisis and fighting for common-sense gun safety,” Ms. Raimondo said.