He has been endorsed by dozens of Democratic politicians who have benefited from his spending. A number of them are members of Congress, but most are mayors, from cities like Houston, San Francisco, San Jose and Washington.

Win or lose, Mr. Bloomberg is the Democrats’ most important donor

Mr. Bloomberg has given tens of millions of dollars to congressional candidates, helping Democrats seize the House in 2018. But in addition to his overtly political spending, he is a leading donor for many top left-wing priorities, including gun control and climate change, supporting organizations like the Sierra Club and Planned Parenthood.

That has created a complicated set of incentives for Democratic groups that either work with Mr. Bloomberg already or hope to work with him in the future. So far, most organizations and politicians that have received Mr. Bloomberg’s money have not endorsed his candidacy, but a number of them acknowledged that they are keenly sensitive to his interests and take pains not to alienate him needlessly.

There is no indication that Mr. Bloomberg has threatened or coerced people in order to get his way. But many people called his more than $60 billion fortune a force powerful enough to make coercion unnecessary.

Mr. Bloomberg represents a test for the Democratic Party

Other candidates like Senator Elizabeth Warren of Massachusetts have accused Mr. Bloomberg of trying to buy the party’s nomination. It remains to be seen whether self-funding his campaign to the tune of $400 million and counting can get him past the likes of Senator Bernie Sanders of Vermont or fellow former Mayor Pete Buttigieg of South Bend, Ind.

His campaign has already proved that many party stalwarts will hold back criticism of stances and statements that typically trouble Democrats, including Mr. Bloomberg’s support for stop-and-frisk policing, charter schools and big banks, as well as his past skepticism about the #MeToo movement and crude comments on women.

Mr. Bloomberg has said he will use his fortune to defeat President Trump in 2020, no matter who the Democratic nominee is. But his campaign has indicated that he may spend more broadly if the party chooses him.