A new front opened on Wednesday in an escalating battle among Democrats over how to handle large numbers of immigrants crossing the southern border and moving into major cities.

The leaders of New York City and New York State, where officials say the arrival of migrants has set off a humanitarian crisis, seemed to turn on each other this week, after the state sent a scathing letter accusing the city of resisting its help and being slow to act.

In the letter Tuesday night, a lawyer representing Gov. Kathy Hochul faulted Mayor Eric Adams’s management of New York’s migrant crisis in sharp terms, puncturing the appearance of city-state harmony that the two leaders have spent much of their tenures cultivating.

New York City is struggling to accommodate more than 100,000 migrants who have arrived after crossing the border, more than 57,000 of whom remain in city shelters. Mr. Adams has said that the city is running out of space and funds to support them, and has criticized President Biden, saying “the president and the White House have failed New York City on this issue.” His posture has infuriated top Biden aides.

Republicans have already homed in on the migrant crisis to attack Democrats ahead of the 2024 elections. Ms. Hochul has been cautious about taking any actions — such as executive orders to compel unwilling counties to take in migrants — that could lead them to amplify those attacks.

So far, Mr. Adams and other Democrats have focused much of their fire on the White House. The state’s letter could disturb that dynamic.

In an apparent attempt to fend off even costlier new requests from the Adams administration, the letter criticizes the city for failing to accept numerous state offers of assistance over the last year, including the use of more than a dozen state-controlled sites that it said could be converted into shelters to house more than 3,000 migrants.

It blames the city for being slow to act, saying the Adams administration ignored a suggestion to begin setting up large “tent-based or base camp sites” for single adult men beginning in June 2022, waiting months to do so. (A spokesman for the governor later clarified that state officials pushed for congregate shelters in July, and tent camps in the fall.)

The letter also says the city did not prioritize helping migrants fill out paperwork to start getting their work permits, meaning thousands who could now be working are not.

“The city has not made timely requests for regulatory changes, has not always promptly shared necessary information with the state, has not implemented programs in a timely manner, and has not consulted the state before taking certain actions,” the state’s lawyer, Faith E. Gay, wrote.

She added: “The city can and should do more to act in a proactive and collaborative manner with the state.”

The 12-page letter, which was obtained by The New York Times, constituted Albany’s formal response to a list of requests the city privately presented to a state judge in Manhattan last week as part of a legal proceeding over how to care for the influx of migrants.

In a separate letter last week, the contents of which have not been previously reported, the city laid out its requests to the state, asking it to cover two-thirds of the cost of sheltering the migrants “in the absence of meaningful federal funding.”

The letter, signed by the city’s assistant corporation counsel, Daniel R. Perez, asked the state to assert more organizational ownership of the problem by implementing “a statewide relocation program to resettle groups of new arrivals throughout the state’s counties in proportion to the counties’ respective share of the state population.”

While migrants have been arriving in New York City in large numbers since last year, their presence has recently grown more visible. It has become commonplace to see migrant women and children selling candy on city subways. And the city’s choices of sites to place shelters, including soccer fields, recreation centers and a parking lot in Queens, have angered some residents and led to protests.

In late July and early August, the city left hundreds of migrants to sleep on the sidewalk outside the Roosevelt Hotel in Midtown Manhattan, which has been turned into a city intake center, even though the city reportedly had hundreds of beds available in shelters.

In response, the Legal Aid Society asked a State Supreme Court in Manhattan to step in to enforce the city’s court-ordered mandate to provide housing to every homeless person who requests it. The judge overseeing the case then ordered the city to identify “the resources and facilities owned, operated and/or controlled by the state” that could help it keep the migrants housed and indicated she would push the state to take a more active role.

The city responded last Wednesday, with the letter requesting the state fill the void left by the federal government, by footing more of the bill and implementing a statewide resettlement plan.

Ms. Hochul’s forceful rebuke was the latest fissure among Democrats in response to the migrant crisis.

For months, New York Democrats, including the governor and the mayor, have blamed President Biden for a weak federal response and have demanded he provide more funding, expedite work permits for migrants and free up federal facilities where they may be housed.

But Ms. Hochul herself has increasingly been the target of criticism, as well. The mayor, as well as community organizations aiding migrants, has called on the governor to take a more hands-on approach to ease New York City’s burden.

“The governor’s response so far has been disturbingly inadequate,” said Dave Giffen, the executive director of Coalition for the Homeless, an advocacy group. “From Day 1, she should have been involved here on the front lines. This is a state issue.”

Avi Small, a spokesman for Ms. Hochul, said Wednesday morning that the governor “is grateful to Mayor Adams and his team for their work to address this ongoing humanitarian crisis.” He added: “Governor Hochul has deployed unprecedented resources to support the City’s efforts and will continue working closely with them to provide aid and support.”

Fabien Levy, the city’s deputy mayor for communications, said the city appreciated the state’s funding promises and its offer of a new shelter site, in the parking lot of the Creedmoor Psychiatric Facility in Queens, which opened yesterday.

“But because this is such a significant crisis, we need more,” he said, “including more space around the state and a statewide order that bars localities from passing ordinances that prevent asylum seekers from being relocated to other parts of New York State.”

Ms. Gay, who wrote the letter sent Tuesday night by the state, is a lawyer at Selendy Gay Elsberg, a city-based firm. Ms. Hochul retained the firm last week after the state attorney general took the unusual step of declining to represent her, citing a “philosophical difference” over the state’s role in managing the crisis.

Ms. Hochul’s response may also reflect another calculation. New York City has a legal obligation to provide a bed to anyone who asks for it. Ms. Hochul indicated last week that she was concerned the judge might extend the “right to shelter” mandate statewide.

The city asked the courts in May to relieve it of its own obligation — so far without success.

In the meantime, the city asked the state in its letter “to consider engaging with neighboring states to arrange for the resettlement of new arrivals outside the state.”

It specifically requested shelters at the Javits Center in Midtown Manhattan, vacant upstate summer camps, shuttered drug treatment facilities and State University of New York dormitories.

In its written response, the state cast itself as a “vital partner” ready to make extraordinary efforts to help the city, but appeared to reject some of the city’s proposals and chastised City Hall for moving too slowly to take advantage of existing legal and financial lifelines.

For example, the state has promised to spend $1.5 billion this fiscal year to, among other things, reimburse the city for shelter costs and provide nearly 2,000 National Guard members to staff migrant sites. The state’s letter said that it had already advanced $250 million of those funds, but had only received reimbursement documentation from the city for $138 million.

The state noted that it was providing the funds, “despite public reporting that raises substantial questions about the operations and conduct of the City’s primary contractor.” The letter refers to a New York Times report about the city’s reliance on a problem-plagued medical services company, DocGo, which won a $432 million no-bid contract.

The state said it had also offered to pay for 1,250 migrant households to move out of the city and fund a year of their rent. But its letter said city officials had given the state a list of only 252 households — just 17 of which “have expressed a willingness to move.”

The city has said the eligibility requirements are too restrictive, and asked that the state ease requirements such as one requiring migrants to complete asylum applications before arriving.

And though the state has offered to let the city use more than a dozen separate state sites in New York City as shelters — including armories, college gyms and the Aqueduct Racetrack in Queens — that together could house more than 3,000 migrants, “the city has not accepted these offers,” the state said.

The governor’s lawyer also suggested that a statewide resettlement plan, the city’s first request, was unfeasible.

“Many migrants will not willingly move outside of the city, and the state will not sanction a policy of involuntarily relocating individuals or families within or beyond state borders,” the Hochul administration’s letter reads.

“As one example of the negative impacts of such efforts, on Aug. 7, 2023, the city sent a bus of 77 individuals to Rochester, New York, and 30 of those individuals refused to get off the bus and returned to the city.”

Andy Newman contributed reporting.