With two days left before lawmakers depart for a seven-week break and shelter-seeking families now being placed on a waitlist, House and Senate Democrats disagree on how prescriptive their crisis response should be.

Senate Democrats indicated Monday they favor putting another $250 million toward soaring shelter costs, the same amount that featured in a spending bill the House approved last week, but the plan they rolled out Monday instead gives the Healey administration more leeway to continue dictating the state’s approach.

The Senate’s supplemental budget requires some of the quarter-billion dollars for shelters to be spent, without specifying any minimum amounts, on wraparound supports for affected families and reimbursements to school districts that have enrolled children who recently arrived in Massachusetts. It also allows — but does not order — the administration to put funding toward temporary shelter sites.

That’s a contrast from the House bill, which would dictate how much money needs to go toward each shelter-related use, including $50 million for creation of at least one “overflow site” that would support eligible families for whom shelter space is not immediately available.

Under the House bill, if the Healey administration fails to launch at least one overflow site within 30 days of the bill’s signing, officials would need to revoke the governor’s order capping shelter capacity at 7,500 families.

Senate Ways and Means Committee Co-chair Michael Rodrigues said his team wanted to give the Healey administration “the ability to adjust on the fly because things are happening so quickly.”

“They can absolutely use these dollars for temporary shelters — however you want to describe them — overflow sites or emergency shelters. But it doesn’t mandate that,” Rodrigues said. “I and we are not experts in how to best handle the migrant crisis situation. We feel that providing the administration the resources, along with strong accountability by requiring robust reporting, is the best way to handle the situation.”

Senators on Tuesday will take up their $2.8 billion spending bill, which also seeks to schedule the 2024 state primary election, clear the way for construction of a professional soccer stadium in Everett and close the state’s financial books for fiscal year 2023.

Rodrigues said he wants to get a compromise bill to Healey’s desk by the time the House and Senate wrap up their formal sessions for the year on Wednesday, a timeframe that won’t allow branch leaders to dig in on their priorities the way they have on other major bills.