GOV. MAURA HEALEY’S call for a cap on the state’s emergency shelter program received tepid support on Monday from the Senate president and House speaker, who said the federal solution the governor is hoping for to address the influx of migrants is unlikely to materialize any time soon because of the leadership void in the US House.

Healey announced last week that she is not ending the right to shelter law, which requires the state to provide emergency shelter to any family or pregnant women in Massachusetts lacking access to housing. But the governor also said she intends to cap the number of participants in the state’s emergency shelter program at 24,000 and place applicants beyond that number on a waiting list.

After a meeting of Beacon Hill’s Big Three, neither Senate President Karen Spilka nor House Speaker Ron Mariano embraced the governor’s approach but neither did they say whether they thought a cap would run afoul of the right to shelter law.

“The concern is that space is at a premium. There just seems to be no more space,” Spilka said. “I believe that options are being looked at right now. We’ll continue to discuss this and work with the administration.”

Mariano indicated he would keep an open mind. “I’m willing to listen to any potential solution that may bring this influx of folks under control,” he said.

But Mariano and Spilka said it was unlikely the federal government would ride to the rescue. “If you’re expecting money to come out of Washington now, you might as well go buy a bridge in New York City. The chances of both are pretty slim,” he said.

Spilka credited President Biden for seeking $1.4 billion in funding to help states like Massachusetts deal with an influx of migrants, but she wasn’t optimistic Congress would act on the request any time soon. “Getting something through Congress right now, you all know the situation in the House,” she said, referring to the failure of the stalemated chamber to elect a new speaker.

Healey said the state has reached its limit in terms of places to house people, in terms of personnel to help them, and in terms of funding. She has a request pending with the Legislature for an extra $250 million for the emergency shelter program, on top of the $325 million authorized in the budget.

“Massachusetts has done its job,” Healey said, indicating the state cannot be expected to go beyond the cap she has proposed. “This is a federal problem. It demands a federal solution.”

Mariano said the top House and Senate budget officials are negotiating over how much additional state funding to provide to the emergency shelter system.

The cost of the program remains somewhat of a mystery. When Healey called a state of emergency to deal with the shelter situation in August, she said at that time that the cost to the state was running at $45 million a month. On Monday, more than two months later, she said the cost has remained the same, even though the number of participants has risen fairly dramatically.

On August 8, when she declared the public emergency, she said there were 5,600 families, and 20,000 individuals, receiving emergency shelter. When she asked the Legislature for $250 million more for the emergency shelter program in mid-September, the number of families had risen to 6,300. Healey said last week the number is expected to rise to 7,500 families / 24,000 individuals by the end of the month.

A Healey spokeswoman said the cost of the emergency shelter program has likely increased since August, but $45 million remains the best estimate because the situation is evolving so quickly. She said the Healey administration expects existing funding to run out in January with no curb on shelter expansion, but by capping the growth at the end of October the administration now expects an extra $250 million from the Legislature would allow the program to continue operating longer.