Top House Democrats proposed spending $250 million to support Massachusetts’ response to an influx of migrants and the creation of an overflow site for families who are placed on a waitlist for emergency shelter once the emergency shelter system hits capacity.

House budget writers were in the process of advancing Tuesday morning a $2.7 billion supplemental budget that included hundreds of millions for the “ongoing humanitarian crisis.” The House plans to vote on the bill during a Wednesday formal session.

The migrant-related dollars matched a funding request Gov. Maura Healey made earlier this year but differed on the details.

The proposal includes more stipulations on specific spending items and even the potential to delay a cap on the number of families in emergency shelters the Healey administration put in place if an overflow site is not created.

House lawmakers include $50 million to create the emergency shelter overflow site for families who are on a waitlist as a result of the system reaching the 7,500 family limit.

The site, or sites, must be operational within 30 days after the spending bill is signed into law or the capacity limit will be “revoked” until the overflow site is “secured and operational,” the bill text said.

Top House Democrats also want the Healey administration to submit monthly reports on the total number of new families in the emergency housing assistance program who entered as migrants, refugees, or asylum seekers; number of families living in hotels and motels broken down by municipality; individuals with work authorizations; and the total amount spent on emergency assistance.

The bill includes $75 million for supplemental school district costs associated with additional student enrollment, $65 million for “sheltering eligible families,” $18 million for temporary emergency shelter sites, and $10 million for resettlement agencies.

House Speaker Ronald Mariano said Monday the chamber would take up a bill Wednesday that included money for emergency shelters but did not immediately disclose a dollar figure.

Healey asked for $250 million with $130 million for sheltering eligible families, $33 million for expanding temporary emergency shelter sites, and $87 million to expand “non-housing specific services at shelter sites and community supports.”

The Healey administration has argued the emergency shelter system is on the brink of disaster with funding running out and the budget facing a deficit without a cash infusion. A surge of newly-arrived migrants coupled with crushing housing costs have led to an increased demand for shelter.

Healey has long called on the federal government to step in and help, and asked state lawmakers for more money for the emergency shelter system.

A judge cleared the way last week for a limit to be placed on the number of families in the system, with state lawyers arguing the shelter system has neither the funding, space, nor personnel to expand indefinitely.

Healey has also floated the potential for time limits to be placed on families’ shelter stays, a possibility that has drawn mixed reactions among providers, advocates, and resettlement agencies.