Lawmakers in the State Assembly say there is a way to offer more support for private and parochial schools without agreeing to generous tax credits for those able to make big-ticket donations.

Instead of reimbursing donors who support private-school scholarship funds for low-income students, the lawmakers said existing reimbursement programs within the state education department could be used to funnel more money to nonpublic schools. The idea would be coupled with a tax deduction for middle- and low-income families that spend up to $3,000 on education expenses for their children each year, and came up during closed-door meetings between Democrats in the Assembly as a counterproposal in contentious negotiations.

“It’s a compromise,” said David Weprin, an Assembly member from Queens.

The lawmakers mentioned the idea, which they cautioned may not yet be on the table among legislative leaders, as they prepared to file out of Albany on Friday. The legislative session was scheduled to end on Wednesday but has been extended until at least next Tuesday, and lawmakers have come to no resolution on a host of outstanding education issues — including mayoral control of city schools, charter-school growth, and revisions to an unpopular teacher-evaluation law — that have been preempted by more urgent negotiations over housing laws.

The most the legislature has been able to agree to so far has been an extension of rent regulations for long enough for lawmakers to go home for a couple of days.

On Friday, Assembly Democrats dropped a months-long call for stronger rent laws and introduced legislation that would simply renew for two years the regulations that expired earlier this week. The bill, first reported by the Daily News, also reintroduces the Assembly’s three-year renewal of mayoral control, conceding nothing to the Senate’s one-year proposal that includes perks for charter schools.

“The Assembly Democratic conference will not be held hostage by the Senate,” Assembly spokesman Michael Whyland told the Daily News. “These are important issues for millions of New York City residents and residents throughout the state.”

Negotiations among the state’s leaders will continue through the weekend, and the rest of the legislature will return Tuesday to resume their work. On Friday, Speaker Carl Heastie said that almost everything was still on the table, adding somewhat begrudgingly that even the prospect of allowing more charter schools to open in New York City was being discussed.

“For the most part, the Assembly conference is not big supporters of charter schools,” Heastie said in brief remarks to reporters. “Charters are something that Senate Republicans like to support. They never want them in their districts. I don’t get that one. But that’s something they may have requested.”

“As long as the clock hasn’t run out on the session, I guess a lot of things are on the table,” Heastie said.

When lawmakers return on Tuesday, there will be just a week left before mayoral control expires.

“We can’t let it lapse,” said Linda Rosenthal, an Assembly member who represents the Upper West Side.

“The fact is that the upstate senators — what is it to them to extend mayoral control? Why is that a bad thing?” Rosenthal said. “It’s clearly another issue they want to take the city hostage on.”

Some of Rosenthal’s Democratic colleagues, however, are more ambivalent — or simply opposed.

“You know, back in the days, when the oft-criticized community boards ran education, we had some wonderful districts,” said Michael Benedetto, a Bronx Assembly member and former teacher for nearly three decades at P.S. 160 in Co-op City. (Benedetto voted in favor of the Assembly’s three-year mayoral control extension.)

“Having school boards isn’t fool proof, but the community needs some say in how their children are being educated, as opposed to one person having control over billions of dollars,” said Latrice Walker, an Assembly member from East New York and Brownsville who was one of the few Democrats who voted against the Assembly’s mayoral control bill last month.

Lawmakers in the Assembly said they hoped to find some common ground next year on how to support private and parochial schools. Gov. Andrew Cuomo has aggressively pushed for a set of tax credits in a bid to help nonpublic schools, which he says would offer needed help to Catholic and other schools. Teachers unions and many lawmakers oppose the plan, which they see a way to direct taxpayer dollars away from public schools. They have also bristled at a tax policy that would provide a 75 percent reimbursement to donors who give up to $1 million, a scheme they see as a giveaway to the rich.

The compromise that the Assembly has floated would boost the amount provided to nonpublic schools through the state’s largest reimbursement program, called mandated services aid. The program, first established in 1974, and others like it reimburse schools for costs associated with reporting daily attendance, administering state tests, giving immunization shots and buying textbooks and technology. Last year, the city’s budget included $64.6 million for nonpublic schools, according to the Department of Education website.

But the programs have not been fully funded for much of the last decade and owes nonpublic schools an estimated $225 million in “delinquent reimbursements,” according to the New York State Catholic Conference last year.

Lawmakers said they preferred using an existing funding stream instead of establishing an entirely new one, as the tax credits would require.

“We’re already doing mandated services and these are resources that private schools have access to,” Walker said.

The state set aside more than $170 million in money to reimburse nonpublic schools this year, Assembly member Catherine Nolan said. Reimbursement money on top of that, the exact amount of which Weprin said was still a moving target, wouldn’t be earmarked to serve low-income families the way the tax credits are, but schools could use it for similar purposes.

“The schools would be in a better financial position which would enable them to potentially reduce tuition, potentially have more scholarship money for students,” Weprin said. “The money is fungible.”

Representatives for the governor’s office and the Senate did not respond to requests for comment.

Bob Bellafiore, a spokesman for the Coalition for Opportunity in Education, which has aggressively lobbied in support of the tax credits, criticized the alternative proposal because it would not include as much funding for needy families.

“For poor and middle-income parents, you need to have the same access to choice that wealthy people have,” Bellafiore said. “Throwing some peanuts to some schools is not going to do that.”