Massachusetts could begin turning away unhoused families next week, and the state’s top elected officials did not seem to have an understanding Monday about what will happen once the state shelter system reaches capacity.

Gov. Maura Healey, Senate President Karen Spilka and House Speaker Ron Mariano were in agreement after a private sit-down that the state’s shelter system is being pushed to its limit by a steady influx of immigrant families arriving from other countries and that something needs to be done. But firm answers about what will happen to families in need of shelter and whether Massachusetts will be violating its right-to-shelter law if it turns a family away were elusive.

“Well, I know that we’re looking with it. We had some discussions about it. The concern is that the space is at a premium, there just seems to be no more space and I believe that options are being looked at right now. So we’ll be continuing discussing this and working with the administration,” Spilka said.

Asked about “legislative options,” the president said, “We’re open to possibilities. We haven’t fully discussed that though.”

Spilka said Beacon Hill leaders at their Monday afternoon meeting discussed the question of what will happen to the first family that Massachusetts turns away. But, she said, “There’s still, details are being worked out.”

Mariano said his impression “is we have to do something” and said he is “willing to listen to any potential solution that may bring this influx of folks under control.”

Healey’s office said it will provide more details before Nov. 1 about what will happen to families turned away from shelter. But initially, it said, families will be assessed for emergency assistance eligibility and will receive a health and personal safety screening. If there is no capacity that night in the state shelter system, the Healey administration said families “will be placed on a waitlist maintained by our administration and will be provided information about shelter alternatives and community resources.”

Neither Spilka nor Mariano said whether they believe it is legal for Massachusetts, which is the only state in the nation with a statutory obligation to provide shelter to unhoused families and pregnant women, to turn people away from the emergency shelter system as Healey last week announced the state will do as soon as the end of the month.

Last week, Healey said about half of the more than 23,000 individuals in the state’s shelter system are new immigrants. She added that children make up about half of the state shelter population, which included about 7,000 families. She said the state will only be able to support 7,500 families and that she expects “to hit that limit at the end of the month.”

“The fact of the matter is we have reached our limit with capacity — the physical infrastructure, places to house people. We’ve reached our limit with personnel — the number of service providers,” Healey said Monday. She added, “So we’ve reached capacity when it comes to infrastructure, when it comes to personnel, and when it comes to funding — you’re all well aware of the amount of money that has been expended to care for new arrivals. While we put forward more additional money through a supplemental budget proposal, that’ll be the subject of discussion. But I think the important point here is that Massachusetts has done its job and so many have come together to make that possible.”

Healey requested $250 million in a supplemental budget bill to help manage the shelter system crisis in mid-September. Mariano has repeatedly said the House wants more information about the actual costs of sheltering migrant families before it can act on the governor’s request. The state is still spending about $45 million per month to shelter immigrants, the governor said Monday.

Asked whether the migrant money is holding up that supplemental budget — it is the fiscal year 2023 closeout budget, which previous state comptrollers have insisted should be done in September to allow them to meet an Oct. 31 deadline to file a key annual financial report — Mariano said the House has “a couple of options around the migrant money.”

“We can separate it out and still do a supp on the close-out,” he said. “It’s in the process of negotiations, but it’s not holding up the supp.”

Mariano clarified that House Ways and Means Chairman Aaron Michlewitz and Senate Ways and Means Chairman Michael Rodrigues have been negotiating the terms of a final supplemental budget before either branch has considered it. Generally, the House and Senate each pass their own versions of a bill and iron out the differences after the fact.

The speaker said the branches are taking the pre-conference route “because it’d be far more efficient to do that and we would be able to take care of some items that need to be taken care of immediately.”

Mariano said the House does not have a timeline for its consideration of the supplemental budget.

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