BOSTON — With the state’s emergency shelters on the verge of hitting a new capacity limit, House Democrats on Tuesday unveiled a new plan to create an “overflow” option that would support families for whom space is not immediately available.

Venturing into the shelter crisis after months of effectively letting Gov. Maura Healey call the shots, House Democrats are newly flexing their legislative muscle by seeking to prevent the governor from capping the number of families in shelters unless she also stands up some kind of alternative for those stuck on the waitlist she plans to deploy.

The House Ways and Means Committee released a long-awaited $2.74 billion spending bill (H 4167) that combines new shelter funding and policies with a bevy of appropriations, including the action needed to close the financial books on fiscal year 2023.

Representatives will be asked to vote on the spending bill on Wednesday, one week before both branches must conclude formal sessions for the year under legislative rules.

The bill would authorize the full $250 million injection into the shelter system that Healey requested nearly two months ago while setting additional requirements on how the money can be used.

One of the most significant measures would take aim at the 7,500-family capacity limit the administration set by requiring $50 million of the $250 million to go toward “the identification, acquisition and operationalization of a state funded overflow emergency shelter site or sites” for families placed on a shelter waitlist.

Under the House proposal, the administration would have 30 days after the effective date of the legislation to launch an overflow site. If officials fail to do so, the bill would require Housing Secretary Ed Augustus to revoke the limit on how many families can be in the shelter system “until said overflow site or sites are secured and operational.”

It was not immediately clear Tuesday — the same day the administration announced a $5 million grant program with the United Way of Massachusetts Bay to support overnight “safety net” shelters — what kinds of services the proposed overflow sites would provide.

The bill also calls for the administration to provide a 60-day notice if officials want to cap how long a family could stay in emergency assistance shelter, a step Healey said last week is under consideration.

Anti-homelessness advocates who have been pressing the Legislature to get involved in the emergency response praised the House’s legislation. Kelly Turley, associate director of the Massachusetts Coalition for the Homeless, said the bill would “address some of the gaps we found” in the Healey administration’s waitlist plan.

“We know without intervention, families will be put on a waiting list and left without adequate shelter in the meantime,” she said.

Out of the $250 million for emergency assistance shelters, the House bill would also require $75 million go toward reimbursing school districts for the costs of enrolling new students who recently arrived, $18 million for temporary shelters, $12 million for clinical and wraparound services, $10 million for resettlement agencies, $6 million for municipal support, $6 million for shelter staffing needs, $5 million for workforce authorization programming and $3 million for family welcome centers, according to a bill summary.

Healey first signaled plans in mid-October to limit shelter capacity at 7,500 families, warning that the system could not accommodate such an unprecedented increase in demand fueled in part by newly arriving migrants.

A Superior Court judge ruled last week that the cap can move forward without running afoul of a 1983 state law guaranteeing shelter to eligible families and pregnant women.

Once the number of families in the system reaches 7,500, eligible new applicants will instead be placed on a waitlist, prioritized based on factors such as whether they have an infant or face risks of domestic violence.

The House proposal seeks to more clearly offer some kind of stopgap support to families for whom shelter space is not available, but it appears likely that at least some people will spend time on a waitlist — with all the uncertainty that entails — without the option of an overflow site.

Healey expects to cross the threshold of 7,500 families in the emergency shelter system this week. Even if the House’s proposal quickly sails into law, the administration would get 30 days to stand up overflow sites before the cap might be affected. And stretching the timeline further, there’s no indication if the Senate also supports the idea, or how feasible the Healey administration thinks it is.

Representatives plan to take up the bill Wednesday, and there’s no clear timeline for action in the Senate, though both branches need to complete formal sessions for the year by Nov. 15. If the bill is enacted as drafted, the administration would get another 30 days to stand up overflow sites before the capacity limit is affected. The first day of winter is Dec. 21.

“Requiring a commitment to a date certain with the overflow sites is very important,” said Andrea Park, director of community driven advocacy at the Massachusetts Law Reform Institute. “There’s still some time in between where families are potentially going to be left without any place to go, but requiring the administration to work toward getting those overflow sites set up will help mitigate some of the very real damage that can happen to families in the winter months in particular.”

Lawyers for Civil Rights, the group that sued the Healey administration over its waitlist plan, said Tuesday it plans to dismiss its so-far-unsuccessful lawsuit in the coming days because the House bill has “achieved the goals of our lawsuit.”

“We are proud that the lawsuit we filed late last month helped bring about this result. Our lawsuit highlighted the plight of families and pregnant women on the brink of homelessness, particularly as winter is approaching,” the group said in a statement. “The lawsuit focused on giving the Legislature adequate time and information to evaluate the need for additional funding. In response to our lawsuit, the State issued emergency regulations on October 31, 2023, for the first time outlining exactly what changes it planned for the emergency shelter program. With that information in hand, the Legislature has responded quickly with additional funding and with new provisions to support families placed on the waitlist.”

While the emergency shelter funding and policies take aim at one of the most pressing crises facing state government, it also represents less than a tenth of the bill’s bottom line.

Most of the money, about $2.1 billion, would go toward MassHealth for what the House Ways and Means Committee called “caseload adjustments.”

Healey proposed that funding in a September supplemental budget. Administration officials have said the MassHealth funding effectively proposes to make payments early to smooth out the expiration of pandemic-era expanded federal reimbursements. By making what would normally be FY24 payments in the FY23 closeout, the impact from those smaller reimbursements would not be as sharp or limited to a single fiscal year, officials said.

In addition, the spending bill includes nearly $300 million for a reserve to fund collective bargaining agreements with state employees. In Healey’s original supplemental budget filed in September, she called for $200 million to go into the fund.

Thousands of state employees are awaiting those dollars, which will fulfill dozens of contracts that have, for some, have been ratified for months. Human service workers and educators who have been waiting for the Legislature to create the funding for these raises staged a rally at the State House last month.

“They were owed this five months ago, and they’re upset the raises haven’t gone through yet. They are hurting,” SEIU Local 509 President Dave Foley said at the time.

The contracts cover employees at the Department of Children and Families, Department of Transitional Assistance, Department of Developmental Services, Department of Mental Health, MassDOT, University of Massachusetts, and more.

The spending bill also includes $10 million in additional flood relief for municipalities, after severe rain events this summer impacted farms and infrastructure across western and central Massachusetts.

House budget writers also picked up the loose ends of a July supplemental budget — including $100 million for pension obligations related to an early retirement program, $75 million for special education reimbursements and $60 million for a DTA caseworker reserve — that didn’t make it across the finish line previously.

Public school districts have been sounding an alarm over increased special education costs after pandemic-era federal grant dollars expired. Some communities have said they will have to pay hundreds of thousands of dollars, or even a couple million dollars, more this year to serve the same amount of students, due to a sizable increase in tuition that schools pay to special education providers.

Healey and lawmakers have steered some money to districts facing this lofty bill in previous supplemental spending and the fiscal 2024 budget, but the House’s plan to direct the large $75 million amount to special education reimbursement was left on the cutting room floor in July.

The House bill also takes another swing at language that would allow the state to renegotiate a hydropower transmission project to try to breathe life back into a stalled renewable energy proposal.

The language, which was included in the House version of the July spending bill, would open the door for Avangrid and electric distribution companies to renegotiate part of their 2019 contract to bring hydropower to New England from a generating station in Quebec.

The renegotiations would likely mean that savings for Massachusetts utility ratepayers would not be as large as originally projected, but the process would help revive momentum for the New England Clean Energy Connect proposal.

The hydroelectric power that NECEC would bring represents about 18 percent of the total energy load in Massachusetts, according to the Department of Energy Resources. Some lawmakers have looked to advance hydropower as the state’s nascent offshore wind industry struggles to get off the ground and deadlines for emissions reductions targets inch closer.

A House Ways and Means Committee spokesperson said the bill would authorize spending from both the state’s general fund and a “transitional escrow fund” full of one-time relief dollars. The legislation empowers the administration to decide how much money to draw from each source, the spokesperson said.