Two veteran Democrats are challenging Mr. de Blasio in the primary: Tony Avella, a state senator from Queens who has challenged the mayor’s approach to homelessness and fought for a Staten Island family trying to keep a large pig as a pet; and Sal Albanese, an Italian-born lawyer and former City Councilman from southwestern Brooklyn who, in his third mayoral run, has argued for a new tax on nonprimary residences — a “pied-à-terre” tax — to help fund affordable housing.

“There’s a lot of anti-de Blasio feeling out there,” Mr. Albanese said.

“No matter where I go, people say, ‘It’s great that you’re running,’” Mr. Avella said.

And, as of Thursday, there was suddenly Roque De La Fuente, a millionaire developer from San Diego who ran for president in 2016 as a Democrat but may run as a Republican in New York. Why does he want to be mayor? “He wants to be president of the United States,” his campaign manager, Rey Olsen, said. New York City mayor is arguably the next best job in politics, he added.

Mr. De La Fuente’s desire to run may be real, but his first problem is that he does not live here. Mr. De La Fuente, also known as Rocky, was rejected by the management of an upscale Fifth Avenue hotel and apartment building, the Sherry-Netherland, after he tried to buy an apartment there, his campaign said. If elected, he would have to establish residency in the city before election day.

Mr. Olsen said that the building management had “refused to interview” Mr. De La Fuente because he is Hispanic (his parents were born in Mexico), and that the candidate had vowed to sue in federal court. Michael Ullman, an executive vice president at the Sherry-Netherland, declined to comment.

The prospect of a well-heeled new candidate entering the Republican race could further complicate efforts by party leaders to avoid a primary and focus attention on Mr. de Blasio.