Bloomberg May Run for President as a Democrat. His Views on Policing and #MeToo Could Be a Problem.

“He’s still a Republican, isn’t he?” Ms. Buckley said at a fund-raising dinner hosted by the Women’s Democratic Club of Clark County. Of Mr. Bloomberg running as a Democrat, she said, “I think people would question why he’s changing at this point in his career.”

Tick Segerblom, a progressive lawmaker in Nevada, said he appreciated Mr. Bloomberg as an ally of the Democratic Party and would keep an open mind about him as a candidate. Mr. Segerblom, who hosted Ms. Warren at an event over the summer, volunteered to welcome Mr. Bloomberg at his home.

“He’s been so fantastic on the environment and so fantastic on guns,” Mr. Segerblom said. “I don’t know, when you get into some of the economic issues, how progressive he is.”

Mr. Bloomberg’s advertising for House Democrats is expected to begin in the coming days, with his spending trained on a few clusters of races in expensive television markets, including in California and Pennsylvania. His first three targets are Los Angeles-area seats held by Representatives Steve Knight and Dana Rohrabacher, Republicans running for re-election, and an open seat near San Diego held by Representative Darrell Issa, a Republican who is retiring.

The advertising blitz includes $4 million in the final 10 days of the election in the Los Angeles media market alone, aides said. But underscoring Mr. Bloomberg’s discomfort with important elements of the Democratic Party, it is not expected to include California’s 45th Congressional District, where Katie Porter, a liberal law professor who is a protégée of Ms. Warren, is challenging Representative Mimi Walters, a conservative Republican.

Close allies of Mr. Bloomberg are divided as to whether it would be wise for him to run for president in 2020, and at least one longtime associate has predicted that he will never seek the White House. Bradley Tusk, Mr. Bloomberg’s former campaign manager who helped him explore an independent candidacy in 2016, declared at a recent dinner in Washington, D.C., that he expected Mr. Bloomberg to toy with running before opting out yet again, multiple people who attended the event confirmed.

Asked about that prediction, Mr. Tusk said in a text message, “No one is better suited to be president than Mike Bloomberg.”

“Running for president and being president aren’t always the same thing,” Mr. Tusk continued. “So we’ll see what he decides, but he’s the best option by far.”