BOSTON (AP) — Massachusetts shelters are being pushed past their capacity, running out of beds for families, including migrants arriving from other states and residents weathering a housing crunch right before winter, said Democratic Gov. Maura Healey.

Healey has said the state will start a waitlist when the number of families in emergency shelters reaches 7,500. Wednesday’s tally showed 7,488, but at least one immigrant aid organization — La Colaborativa in Chelsea, Massachusetts — said late Wednesday that it appeared the cap had taken effect as it tried to place a family in a shelter.

“We were informed we hit the cap and due to the cap, families are being put on a waitlist,” said Cherlin Dubon, triage case specialist for the group.

Healey has said she doesn’t want to see families out on the street but that the state has essentially reached its shelter capacity. The spike in demand is being driven in part by migrant families entering the state.

Many of the migrants are arriving from other states. Some states led by Republicans — including Texas and Florida — have bused or flown immigrants to states and cities led by Democrats, including California, Massachusetts, New York and Chicago.

New York City Mayor Eric Adams has announced he is limiting shelter stays for migrant families with children to 60 days. In Chicago, officials have looked to relocate migrants seeking asylum from police stations and the city’s airports to winterized camps with massive tents.

Critics argue Healey’s decision to cap shelter placements violates the state’s “right-to-shelter” law. Under the four-decade-old law, Massachusetts is legally required to provide emergency shelter to eligible families.

Under Healey’s plan, women, young children and those with acute medical needs and health issues will be given priority. The state is also considering limiting how long a family can stay in a shelter, Healey said.

With winter not far off, officials are scrambling to prevent families from ending up on the street. On Tuesday, Healey announced a $5 million grant program to help local organizations create overnight shelter for families and pregnant individuals with no other options. Healey has also said she’s pressing federal officials to speed up the process by which migrants can get work authorizations and ultimately exit the shelter system to free up more space.

Massachusetts lawmakers are also weighing a bill to set aside $50 million to set up one or more locations where homeless families could find temporary refuge while they wait for a shelter space.

Democratic House Speaker Ronald Mariano said that could be a single large site like the Hynes Convention Center in Boston or smaller sites spread around the state.

“Where are these people going to go?” Mariano said Wednesday. “Where do they spend the night when they come in here on a Friday night at 7 o’clock?”

For families denied shelter, the state has made a flyer that suggests a handful of options, the first being to “return to the last safe place you stayed.”

Denying families emergency shelter could force some into unsafe living conditions, said Kelly Turley, director of the Massachusetts Coalition for the Homeless.

She and other homeless advocates have pressed the Legislature to approve money for a large living site similar to what Mariano described.

“We’re very concerned that after 40 years of having the right to shelter, that the administration is moving forward with their plan without making sure congregate shelter is available,” Turley said.

Advocates welcoming new migrants to the state say they’re concerned about how to help those with no friends or family and nowhere to stay.

“When people come, especially those with babies, do we send them to the street?” said Geralde Gabeau, executive director of the Immigrant Family Services Institute in Boston. “We are not sending them to the street, so we need a place to send them.”

Families are housed in hundreds of locations in 90 cities and towns in a range of facilities, from traditional shelters to temporary sites like college dorms.

The state Executive Office of Housing and Livable Communities last week issued guidance on the coming changes to the shelter system.

Top priority will be given to families at imminent risk of domestic violence or who have an infant up to 3 months old, have family members with an immunocompromised condition, are experiencing a high-risk pregnancy or who include a family member with a medical device, specifically a tracheostomy tube. Additional priority levels will take into account the age and medical needs of family members.

Under the guidance, families will be offered available shelter units based on their position on the waitlist. The list will be refreshed once a day and those eligible for shelter will be contacted by email, phone call and text. Families on the waitlist for six months or longer will have to undergo another assessment.