Speaking recently to the city’s power brokers at the Real Estate Board of New York’s annual gala, Mayor Eric Adams tried to make the case that his administration had conquered some of the city’s most vexing problems.

He had swept homeless encampments off the street and flooded the subways with police officers. Tourists were returning and the city’s bond rating had improved. But there remained one nagging and growing problem that has so far evaded solution: the influx of more than 59,000 migrants since last spring.

“Remove the $4.6 billion problem we have in asylum seekers,” Mr. Adams said, “and you will see one of the best administrations in budgeting taxpayer dollars in the history of this city.”

In his 16 months as mayor, Mr. Adams has become known for latching onto a particular issue and using it to propel his agenda. His constant focus on shootings and crime, and even on the city’s ubiquitous presence of rats, have become symbols of his fight against disorder.

Unlike criminals and rats, however, the migrants seeking asylum and shelter are not typical targets of vilification, at least not for New York Democrats like Mr. Adams. So in recent weeks, the mayor has tried to carefully toe a rhetorical tightrope: He clearly wants to press his case that the influx of asylum seekers may cripple the city unless the federal government commits more money and focuses more attention to the situation. But he also seems sensitive to not overtly stigmatize those seeking asylum.

During a 15-minute speech last Wednesday, while he was unveiling his $106.7 billion executive budget, the mayor repeatedly referenced the influx of migrants and their effect on city spending.

“The issue is not the asylum seekers, the issue is the fact that the national government is not doing its job,” Mr. Adams said at a news conference following the speech. “And so, if people want to play word police and don’t know my heart by now and what I have done better than any other municipality, then let them do that.”

Still, during a trip to Washington, D.C., the prior week, the mayor said that the city was being “destroyed by the migrant crisis.” And earlier, during a news conference where he criticized President Biden for not doing enough to grant migrants work authorization, Mr. Adams said “every service in the city is going to be impacted by the asylum seeker crisis.”

The city projects spending $4.3 billion over the next two fiscal years to cover the costs of the migrant influx, with only roughly 37 percent expected to be covered by the state and federal government.

“Cuts are coming, and they’re going to blame the newcomers and say that they’re draining resources,” said Hildalyn Colon-Hernandez, deputy director of New Immigrant Community Empowerment, a nonprofit that supports immigrant workers. “You cannot welcome and close the door at the same time. You have to decide.”

At the same time, the asylum seeker costs aren’t the only financial issues the city is facing. The executive budget is $4 billion more than the preliminary budget in January. The city also expects to spend $16 billion to settle its outstanding union contracts and is projecting budget gaps of $4 billion to $7 billion in each of the next three fiscal years. While tax revenue is $4.4 billion more than expected over this year and next, federal pandemic aid is also expiring.

“We would have a significant budget problem to solve even if we weren’t facing the migrant crisis,” said Andrew Rein, president of the nonpartisan Citizens Budget Commission. “We can’t solve the city’s fiscal crisis by only focusing on one part of it.”

Fabien Levy, a spokesman for Mr. Adams, said that the mayor merely wanted to ensure that New York City residents “are not left to shoulder the costs alone” of the migrant influx.

When migrants first began arriving in the city in large numbers, some of them bused here by the governors of Texas and Arizona, the mayor seemingly questioned the city’s unique “right to shelter” law, which requires housing to be provided for anyone who asks.

At the time, the city was already experiencing a rise in the shelter population related to staffing shortages and delays in processing housing applications. The vacancy rate in the city’s family shelters had dipped lower than normal, alarming advocates.

Brad Lander, the city comptroller, said the city deserved credit for the scope of its effort to help more than 59,000 migrants, about 37,000 who are still under the city’s care. Mr. Adams projected that the figure could swell to 70,000 by June of 2024.

The mayor is also worried that the pending expiration of Title 42, a public-health rule that allowed the Trump and Biden administrations to prevent migrants from seeking asylum at the border and to expel them, will mean a new flood of asylum seekers.

City officials said that they had been warned by the federal government to expect the busing of migrants from the southern border in Texas to the city to restart. And on Monday, Mr. Adams said Gov. Greg Abbott of Texas would be playing “politics with people’s lives” should he resume the “uncoordinated and involuntary” busing of migrants to several cities, including New York.

In mid-April, Mr. Adams unleashed his harshest criticism yet against Mr. Biden, an ally who visited New York City last year, for not doing more to approve work authorizations. The Biden administration has called on Congress to enact immigration reform legislation but has also launched efforts to speed work authorization for some asylum seekers.

The city has also yet to change its own rules that would allow shelter residents to move to permanent housing without a 90-day shelter stay, in spite of promising a shift in that direction. The City Council has introduced legislation to waive the rule. Among other legislation being considered by the City Council is an immigrant workers’ bill of rights, creating an office of refugee and migrant resettlement and expanding the list of documents that can be used to secure a city identification.

While Mr. Lander agrees with the mayor that the state and federal governments should contribute more than the $1.6 billion the city is anticipating so far, and that providing migrants with work authorization is a priority, he believes the city can use its resources more wisely.

His office estimates that 99 percent of the spending on migrants is going to emergency shelters. The city has opened more than 100 emergency shelters and eight Humanitarian Emergency Response and Relief Centers. Mr. Lander called on the administration to invest more money to help migrants file the necessary paperwork to begin working and leave shelter.

“If we’re not successful at helping families file their applications and get work authorization, then they’re much more likely to stay in shelter longer and the cost for the city will then be much higher,” Mr. Lander said in an interview.

At the Jackson Heights office of New Immigrant Community Empowerment in Queens, dozens of asylum seekers sat through an orientation last week about how to navigate New York City and do things like get a state identification and find work. Since January, the number of participants has tripled to 1,000 per month.

Nilbia Coyote, the executive director, said the asylum seekers arriving in the city today have skills that could be put to immediate use, but her group is struggling to keep up with new arrivals. A line of people waited outside the group’s storefront office.

“We see this as investing in our city,” Ms. Coyote said about the overflowing orientations. “These folks are here to contribute to the city.”