Divisions among New York Democrats widened on Thursday over the influx of migrants arriving from the southern border, with the state attorney general declining to represent Gov. Kathy Hochul in legal proceedings over how to care for thousands of newcomers.

State officials notified a judge in a public filing in State Supreme Court in Manhattan that a private law firm, Selendy Gay Elsberg, would be taking over the case from Letitia James, the attorney general.

While Ms. James did not immediately comment on the switch, a person familiar with her thinking who was not authorized to discuss it publicly said the attorney general was motivated by fundamental policy disagreements with the governor on the migrant issue.

It was unclear what specific issues Ms. James, who is to Ms. Hochul’s left, had with the governor’s stance on the influx of asylum seekers.

The decision to withdraw amounted to a remarkable break between two of the state’s most powerful Democrats. Though their politics do not always align, the attorney general traditionally acts as the governor’s lawyer in court. Only rarely have elected officials in that position turned down the duty.

The case in question has to do with a decades-old mandate that New York City provide shelter to all who ask for it. With thousands of new migrants arriving each week, the city is struggling to meet its responsibility.

Mayor Eric Adams has repeatedly urged the federal government to immediately authorize the migrants to work legally and to send more money to help house and care for them, which the city estimates could cost as much as $12 billion.

Tom Perez, a senior adviser to President Biden, traveled to New York City on Thursday to meet with state and city officials about their response.

At the same time, Mr. Adams has pushed Ms. Hochul to provide more assistance and develop a plan to distribute arriving migrants across the state.

The governor has already provided $1 billion to assist the city, and promised to seek more funding in next year’s budget and to continue to push Washington for more resources.

But last week, after hundreds of migrants were left to sleep on the sidewalk outside the city’s main intake center, the judge overseeing the case ordered the city to send the state “a proposal identifying the resources and facilities owned, operated and/or controlled by the state” that the city needs to continue providing shelter for everyone who asks.

The city sent its proposal on Wednesday, but it has not been publicly released. The state faces an Aug. 15 court deadline to respond.

A spokesman for Ms. Hochul did not immediately respond to a request for comment.

Ms. James, who, like Mr. Adams, rose through Brooklyn politics, has generally staked out policy positions to the left of Ms. Hochul, a centrist from Buffalo, and has deeper ties to the New York City political class, including the mayor.

Dana Rubinstein contributed reporting.