In a luxury apartment in Trump World Tower in Manhattan near the United Nations, a group of two dozen or so lawyers and civic leaders gathered this spring at the home of Mitchell Draizin, a financier and prominent Democratic donor.

The group, he said, was searching for a candidate to defeat Mayor Eric Adams, “somebody to manage and have the city progress.” And for that evening, Scott Stringer, the city’s former comptroller, had their ear.

“He’s obviously not the most charismatic person. Perhaps Adams is charismatic, but look where it got us,” Mr. Draizin said in an interview. “Look at the M.T.A., look at the garbage, look at the shoplifting and the migrant crisis.”

Similar scenes are playing out across the city, with Democratic supporters being courted by Mr. Adams, Mr. Stringer and others interested in running in next June’s mayoral primary.

While most Democrats are preoccupied with the presidential race this November, the New York City mayoral race has begun and is showing signs of being more competitive than many expected. Mr. Adams, a former police captain who ran on a public safety message, is facing low approval ratings, prompting many Democrats to consider jumping in.

Two Democrats — Mr. Stringer and Zellnor Myrie, a state senator from Brooklyn — have already entered the race, and they announced their fund-raising totals on Monday, the first major reporting deadline. The current city comptroller, Brad Lander, is seriously considering joining the race, along with former Gov. Andrew M. Cuomo.

Mr. Stringer’s campaign said on Monday that it had raised about $424,000 since January from about 2,100 donors, which is expected to translate into as much as $2 million with matching funds.

Mr. Stringer, whose 2021 mayoral campaign was derailed by allegations of sexual misconduct, ran as a progressive last time and did not take donations from the real estate industry. Now he is focusing on arguing that he could run the city better than Mr. Adams and is courting real estate donors.

Mr. Myrie’s campaign said it had raised roughly $326,000 since May from about 1,800 donors, including Jennifer Soros, a member of the wealthy Soros family, and Chris Hughes, a founder of Facebook. That could translate into more than $1.3 million with matching funds.

Vito Pitta, a lawyer for the Adams campaign, said in a statement that the campaign had raised more than $1 million over the last six months. Under the city’s matching funds program, the total fund-raising for the 2025 campaign could grow to more than $8 million, he said, which is close to the maximum allowed.

The mayor’s fund-raising picked up this year after slowing down following an F.B.I. raid in November at the home of his chief fund-raiser. His recent donors include Jonathan Kraft, president of the New England Patriots; several relatives of James Dolan, who oversees a family empire that includes Madison Square Garden and the Knicks; and Robert Alexander, chairman of CBRE, a major commercial real estate firm.

Frank Carone, the mayor’s former chief of staff who plays a leading role in his campaign, said in a statement that the campaign would make a “final push” to finish its fund-raising by the fall so that Mr. Adams could focus on delivering his message to voters ahead of the primary.

As the mayor faces a federal investigation into his campaign fund-raising, his allies have also been raising money for a legal-defense fund that has more than $1 million from donors, including Michael R. Bloomberg, the former mayor.

It is rare for an incumbent Democratic mayor in the city to face a contested primary, especially one involving several candidates. The broad interest in challenging Mr. Adams has created a battle for donors.

John Catsimatidis, the billionaire owner of Gristedes supermarkets and a prominent donor to both parties, said he had decided for now to ignore text solicitations from Mr. Stringer despite their 40-year relationship.

“He reached out, but I’m not going to talk to anybody right now,” Mr. Catsimatidis said. “A, it’s too early. B, I think Eric Adams wants to do a good job, and I think he is hamstrung by other politicians.”

Another donor who is in the real estate field acknowledged holding a recent fund-raiser for Mr. Stringer at a hotel in the NoMad neighborhood of Manhattan. He expressed support for Mr. Stringer’s “technocratic” approach, comparing him to Mayor Fiorello La Guardia, but he asked to remain anonymous over fears of alienating Mr. Adams.

Mr. Myrie, 37, has been racing to introduce himself to voters and to collect small campaign donations to demonstrate a broad grass-roots appeal. He boasts progressive bona fides, influential allies in the State Legislature and a compelling biography as the son of immigrants from Costa Rica.

Still, most New Yorkers do not know his name.

That is how he found himself at the Gin Mill bar on the Upper West Side of Manhattan last week, being feted as “hot Urkel” — a reference to the nerdy neighbor from the 1990s sitcom “Family Matters.” Mr. Myrie’s stump speech about free universal after-school programing appealed to the millennial crowd, and a bartender even asked if he could donate.

“His energy is vibrant and young, but then he has a real gravitas about him, and his legislative accomplishments are real,” said Rachael Bedard, a former doctor at Rikers Island jail complex who attended the event and quit her position on the city’s Board of Correction to raise money for Mr. Myrie’s campaign.

Mr. Adams recently held a fund-raiser at Crown One, a Chinese restaurant in Flushing, Queens, with tickets costing between $250 and $2,100. The event was hosted by his brother Bernard Adams, who served on the mayor’s security detail during his first year in office, and Kaily Cheng, a community board member in Queens.

More than 20 people attended the event and dined on shrimp dumplings and fried rice, Mr. Cheng said in an interview in Cantonese. The mayor shook people’s hands and listened to their concerns, including about crime and the possibility of a new casino near Citi Field.

Mr. Cheng said he had urged the mayor to hire a Chinese police commissioner and was helping to search for potential candidates.

Mr. Adams has significant advantages as an incumbent, including his fund-raising prowess and ability to leverage the powers of the bully pulpit, said Basil Smikle, a professor at the School of Professional Studies at Columbia University.

Mr. Smikle said that Mr. Stringer’s fund-raising haul was a “good number, but I don’t think the mayor is going to look at it and feel threatened.”

As for other Democrats considering entering the race, Mr. Smikle said that he thought Mr. Adams was more vulnerable over his management of the city and not as much over ideology: “There’s a lane for someone to be slightly to the mayor’s left but not too significantly.”

Adena Rosenthal, a school psychologist, held a fund-raiser earlier this year for Mr. Stringer, her friend of 40 years, at her home on the Upper West Side. She served sushi, his favorite food, and invited roughly 20 neighbors and friends. “They are all so excited to have someone else running other than Mayor Adams,” she said.

Mr. Stringer answered questions, including one about the migrant crisis, and pointed out how Mr. Adams had been inconsistent in his approach.

Dr. Rosenthal said that none of the guests appeared concerned about the sexual misconduct allegations that sank Mr. Stringer’s 2021 campaign and that they did not seem legitimate to her: “No one even brought it up to me other than to say it’s a shame he didn’t win.”

Mr. Lander held a “birthday bash” fund-raiser last week at a brewery in Gowanus, Brooklyn. The host committee included Representative Jerrold Nadler and Jumaane Williams, the city’s public advocate, along with progressive donors, many of whom gave $1,000 or more.

The funds will directly benefit Mr. Lander’s re-election campaign for comptroller, but the money could be rolled into a mayoral campaign.

Mr. Lander reported raising just under $200,000 this year, in addition to $470,000 for the 2025 campaign in previous years, and has $156,000 on hand, according to campaign filings. With anticipated matching funds included, that will translate to more than $3 million, his campaign said.