HOUSE LEADERS ON Tuesday indicated they plan to give Gov. Maura Healey the $250 million she asked for in September to support the state’s cash-strapped emergency shelter program, but the money would come with specific directives on how all of it should be spent.

For example, the House Ways and Means legislation would direct the Healey administration to spend $50 million setting up an overflow emergency shelter site or sites for people wait-listed under the governor’s plan, which limits the capacity of the emergency shelter program to 7,500 families.

The proposal also says the overflow site or sites must be operational within 30 days after the bill’s passage or the governor’s emergency shelter declaration capping the program at 7,500 families will be revoked and the cap will be lifted until such time as the overflow center or centers are up and running.

A spokesman for the Ways and Means Committee said an overflow center would be available to families who aren’t immediately placed in the shelter system and end up on a waitlist.

“It would be a site for people who arrive during later hours and weekends when welcome centers or other sites are closed,” the spokesman said. “It would be up to the administration to work out some of the further details on the site or sites. “

Healey has said the state is expected to reach the cap level of 7,500 families shortly and can’t go beyond that level. She said it’s not just a question of money. “We don’t have enough service providers. We don’t have enough physical space,” she said on Monday after a meeting on the shelter crisis with House Speaker Ron Mariano and Senate President Karen Spilka.

The House bill, which is expected to go to the full body on Wednesday, also directs that $75 million go to schools absorbing children residing in shelters, $65 million for shelter services, $18 million for temporary emergency shelter sites, $12 million for wraparound services for families in shelters, $10 million for resettlement agencies, $6 million for shelter staffing, $6 million for municipalities hosting shelters, $5 million for migrant and refugee workforce training programs, and $3 million for the state’s family welcome centers.

All of the $250 million would come out of an account holding budget surpluses from prior years.

The House legislation also requires extensive reporting every 30 days on the costs associated with the state’s emergency shelter program, particularly as it relates to migrants coming to Massachusetts from outside the country. The bill asks for a breakdown of migrant families and asylum seekers entering the shelter program and details on how many of them have work authorizations.

The bill also requires the administration to provide a 60-day notice of any change in benefits under the emergency shelter program, including any bid to limit or adjust “the duration of benefits,” which presumably means a reduction in how long a person can remain in a shelter.

The funding for the emergency shelter system was included in a broader $2.7 billion supplemental spending bill that was released by the Ways and Means Committee at 10 a.m. via email on Tuesday. Members of the committee were given 45 minutes to vote. The House meets in a formal session on Wednesday, where the bill is expected to be approved.

On Monday, late in the afternoon, Mariano said he didn’t know how much money the supplemental spending bill would provide for the emergency shelter system.