The Legal Aid Society sued Mayor Eric Adams on Wednesday for not complying with new laws meant to address rising homelessness and the influx of tens of thousands of migrants.

In the coming days, the City Council is expected to join the society, after voting last week to authorize a lawsuit.

The move comes as a homelessness crisis overwhelms shelters and the city budget and as tensions increase between Mr. Adams and the Council over how to manage it.

The laws at issue are a package of bills the Council passed last year to make more people eligible for a housing voucher program as record numbers of homeless people entered city shelters. Mr. Adams vetoed the legislation, saying it would be too costly.

The Council voted overwhelmingly to override the mayor’s veto in July. But since then, the administration has failed to expand eligibility for the vouchers, according to the lawsuit. While outside groups often sue the city, legal action by the City Council against the administration is more rare.

Legal Aid, a group that represents low-income New Yorkers and others on a variety of legal matters, filed their lawsuit as a class action on behalf of those eligible for the vouchers, known familiarly as CityFHEPs.

“The Adams administration’s refusal to implement the law is unacceptable, and the City must take immediate action to ensure that the thousands of New Yorkers who are experiencing or are on the brink of homelessness and who are now eligible for CityFHEPS can secure safe, long term and affordable housing,” said Robert Desir, staff attorney at Legal Aid.

The voucher program is one of the main ways the city fights homelessness. Eligible renters typically spend no more than 30 percent of their income on rent, with the remainder covered by the city.

The mayor has made some changes to the voucher program by executive order, such as eliminating a rule requiring people to stay in a homeless shelter for 90 days before they are eligible, a change long sought by homelessness advocates.

The package of laws passed by the Council went further, however. Among other things, the laws would make tenants eligible for a voucher after receiving a demand for unpaid rent from their landlord. Previously, people typically had to show they were facing eviction in housing court.

Adrienne Adams, the speaker of the City Council, has said the mayor’s efforts fall short of what the law requires.

“Just like you have to comply with the law, I have to comply with the law, the administration has to comply with the law as well,” Ms. Adams said last week.

Lately, Mr. Adams and the City Council have been at odds over other issues as well. Last month, the Council voted to override Mr. Adams’s vetoes of bills that would require police officers to record the race, age and gender of most people they stop and that would end solitary confinement in the city’s jails.

The mayor has said the changes to the voucher program would cost billions more dollars than the city can afford.

In 2023, the city spent nearly $500 million on the program, almost double what it spent in 2021, according to the city’s Independent Budget Office. The city says vouchers help about 36,000 households with rent.

Tenant and homelessness advocates have argued that moving more people to permanent housing would help ease the burden on the city’s shelter system caused by the influx of migrants over the last two years. More than 66,000 migrants are currently in the city’s care.

During the fiscal year that ended in June 2022 — just before migrant arrivals pushed costs higher — the city shelter system was paid $136 per night or nearly $50,000 per year to house a single adult. It cost $188 per night, or nearly $70,000 per year, to house a family with children.

Christine Quinn, a former City Council speaker and chief executive of WIN, a network of shelters for women and children, said the city could save hundreds of millions of dollars and prevent tens of thousands of people from becoming homeless by implementing the voucher reforms.

Mr. Adams has also said that the city simply does not have enough housing to accommodate everyone with a voucher. The vacancy rate in apartments that rent for less than $2,400 was less than 1 percent in 2023, according to data released by the city last week.

“Idealism collides with realism all the time,” Mr. Adams said at a news conference on Tuesday, adding that increasing the number of voucher holders would only “aggravate” the problem. “The realism is we don’t have the inventory.”

He said the City Council had “good intentions,” but suggested the program could be abused and called for the focus to move to building more housing.

“The way out of this,” Mr. Adams said, “is, No. 1, to build more.”

Pierina Sanchez, a councilwoman from the Bronx who is a sponsor of two pieces of the legislative package, called the mayor’s focus on housing supply shortsighted and his implication that the vouchers might be abused “poverty shaming.”

In her district, where the median income is much lower than the city’s, Ms. Sanchez often has residents who are struggling to stay in their homes.

“We have to do things differently in order to meet the demands of this moment,” Ms. Sanchez said. “Part of doing things differently means expanding access and stabilizing our lowest income New Yorkers.”

Andy Newman contributed reporting.