BOSTON — Less than a week after the state abruptly opened a new temporary shelter for migrant and homeless families in Cambridge, the Healey administration said it has met the Legislature’s new requirement to open an overflow site by the end of the year.

But the House’s top Democrat, who pressed for the overflow site directive, didn’t explicitly agree with that assessment Friday morning.

A spokesperson for Gov. Maura Healey told the News Service that the administration feels that it’s fulfilled the condition she agreed to in a supplemental budget to operationalize by Dec. 31 a “state funded overflow emergency shelter site or sites for eligible families who have been waitlisted for placement at an emergency shelter” due to the system reaching capacity.

The new law doesn’t explicitly define the parameters of an overflow site, such as how many families must be accommodated or how long they may remain at temporary sites.

Asked if he agreed that the overflow site requirement had been met, House Speaker Ron Mariano responded with a statement to the News Service.

“We are hopeful that families on the waitlist are being provided with a safe place to sleep as required by our legislation,” Mariano said. “We will continue to monitor the steps taken to address the shelter crisis, including the required reports, to help ensure that there are operational overflow sites through the end of the fiscal year.”

Some 391 families are on the waitlist, Healey spokesperson Karissa Hand said. Emergency Assistance Director Scott Rice pegged the figure at more than 400 families during a virtual Cambridge community meeting Thursday evening.

The waitlist was at 242 families on Dec. 13. Rice said the average family size is three people.

Throughout the 90-minute virtual forum, Cambridge city and state leaders explained how they selected the Middlesex South Registry of Deeds building and began welcoming families last Friday with little public notice or engagement. About three-quarters of families stuck on the waitlist have been directed to temporary shelter and overnight arrangements, Rice said.

Officials have been scrambling to open temporary overflow sites since the shelter system hit Healey’s 7,500-family capacity limit in November amid a surge of new arrivals, and each day brings about 10 additional families to Massachusetts, Rice said. About five to 10 families are also leaving the shelter system daily, he said.

When Secretary of State William Galvin offered up the east Cambridge property, Rice said his team decided in less than a day to start fixing up the former courthouse.

“We’ve been very fortunate this week — it hasn’t been below freezing very much, but that is a crisis that I’m worried about,” Rice said. “But when I find, and we find as a group and incident command, a location that is worthy of taking a look, we move very rapidly, as rapidly as we can. Do we do it perfectly? Do we have the most perfect community engagement plan? No, we don’t.”

Rice thanked city leaders, including Mayor Sumbul Siddiqui and City Manager Yi-An Huang, for their support and “friendly attitude” in navigating the unpredictable demands of the migrant crisis. Cambridge has been willing to help and “gone above and beyond on trying to solve problems,” Rice said.

“It was not entirely clear that everything was going to be completed in time, even a week before, so a lot of the exploration of the site to make sure that everything was going to be prepared and that it would actually work did happen incredibly fast,” Huang said. “The goal of the administration would be that there is a longer period where this can be noticed to the community and there can be more of a conversation. But then the reality of the emergency and the crisis and finding a place for families, especially with the winter, sometimes it’s not happening as much as we would like.”

The meeting came months after lawmakers criticized the Healey administration’s lack of communication, including with municipalities, as more hotels in their communities began serving as emergency shelters and more migrant children began attending local schools.

In response to Healey’s shelter cap, lawmakers wrote the new law with the overflow requirement in order for the administration to unlock $250 million in additional funding for the emergency shelter system. Healey agreed to the requirement when she signed the supplemental budget that included it on Dec. 4.

The administration must also submit biweekly shelter updates to the House and Senate Ways and Means committees. The first submitted report was dated Dec. 18, and Hand said the administration will submit another report next week.

Rice said the Cambridge Registry of Deeds building is one of five overflow sites, which officials have also referred to as safety-net shelters.

The other state-funded overflow shelters are at Eastern Nazarene College in Quincy and a clinical risk assessment site in Revere, alongside other sites funded through a grant partnership the administration launched with United Way of Massachusetts Bay, Hand said.

“We know that is going to continue being a growing problem before it’s not, and some really hard conversations are going to have to happen,” Rep. Marjorie Decker, a Cambridge Democrat, said of the massive shelter demand. “But I think this kind of collaboration with municipalities across the state, as well as legislators and our state partners is really important.”

The flood of new arrivals is affecting major cities across the country, and New York City Mayor Eric Adams this week pointed the finger at Texas Gov. Greg Abbott for transporting migrants by bus and planes without warning.

Adams on Wednesday issued an executive order stipulating that chartered buses bringing migrants into the city will be required “to provide 32 hours’ notice before arriving in New York City and information on the population they are transporting, as well as be required to drop passengers off at a designated location in Manhattan only during specified hours.”

The mayor’s office said Abbott was using asylum seekers as “political pawns” and noted a surge of 14,700 new arrivals in the last month, including 14 “rogue buses” with migrants that arrived from Texas in a single night.

Violations could result in fines and charter buses being impounded, and Adams signaled city officials may also file lawsuits.

In Cambridge, Rep. Mike Connolly said he’s visited the shelter several times. The initial families assigned to the shelter seemed “quite tired” and “exhausted,” the Cambridge Democrat told the News Service earlier this week.

The space can accommodate up to 200 people, or roughly 60 to 80 families, Rice said.

“These families are mostly migrants, and they’re coming here for the opportunity to pursue the American dream as part of our society,” Rice said. “They want to get out of this system, they want to work. We want to help them get a work authorization as quickly as possible. They’re all here legally, in accordance with the federal government rules and regulations.”

Eligible families on the waitlist are brought to the shelter at 6 p.m. and depart at 7 a.m., said Blair Brown, an assistant education secretary who’s now working on the overflow shelter team. She described the Cambridge site as a “very temporary overnight shelter,” where families are sleeping on cots in two congregate rooms.

Families are provided dinner and breakfast, and MIT has offered shower facilities, Brown said. During the day, she said, the families spend time at the state welcome centers, located in Quincy and Allston, where they can connect with more state resources and determine their next steps.

Maura Pensak, Cambridge’s housing liaison, said it’s better for families to stay in more traditional shelters, where they would have their own space and not need to leave every day.

“This setting is hopefully just a real quick turnaround while they’re waiting,” she said. “Think of it as a waiting room.”