ALBANY — Legislative leaders said Thursday they were focusing on increasing school aid beyond the $1 billion increase already proposed by Gov. Andrew M. Cuomo.

But the closed-door talks stumbled over several issues, including the insistence by Senate Republicans that a cap on charter schools be lifted.

“Huge snag with the Assembly on charters,” said one majority senator familiar with negotiations. The senator spoke on the condition of anonymity because details of talks aren’t supposed to be released.

Assembly Speaker Carl Heastie (D-Bronx) on Wednesday dismissed the Senate’s call to raise cap while there are still dozens of unused charters that private entities could use to open charter schools.

But Assembly Majority Leader Joseph Morelle (D-Rochester) said later Thursday, “Charter schools are in the mix.”

The publicly funded, privately operated schools are limited to 460 schools, compared to 5,000 traditional public schools. Many charter schools report long waiting lists of students, while the state teachers unions and public schools argue that charter schools siphon too much funding from traditional schools.

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The $162 billion budget is due by midnight Friday. Morelle said Assembly members were told to prepare to work in Albany potentially into Saturday, which would make the state budget late. The last two state budgets missed the midnight deadline of March 30 by several hours.

School aid is usually the last issue settled in budget negotiations.

“I think all of us have advocated . . . to increase it above and beyond what the governor did,” said Sen. Jeff Klein (D-Bronx), leader of the Independent Democratic Conference. “We want to do it and I am very optimistic we will.”

Cuomo said Wednesday that he didn’t want to agree to more than his proposed $1 billion increase in state school aid because he fears billions of dollars in likely federal budget cuts.

Heastie wouldn’t comment Thursday after the closed-door negotiating session, but he has advocated for twice the increase in school aid that Cuomo proposed.

“If you look historically,” said Senate Majority Leader John Flanagan (R-East Northport), “the Legislature has generally always added more money to the governor’s [budget proposal]. But the governor has put $1 billion on the table, for which he deserves credit. We may not like the distribution, but he deserves praise for putting that on the table. It is the single largest growth area of the budget.”

In January, Cuomo proposed $24.8 billion in aid to New York’s more than 700 school districts. That would be a 6.5-percent increase in funding, but school officials weren’t pleased that the governor directed $100 million to specific programs and uses including adding pre-Kindergarten classrooms.

As for the overall budget talks, negotiations are headed to the final day of the fiscal year, as is often the case in Albany. The budget is due by midnight Friday.

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The Assembly’s Democratic majority and the Senate’s Republican majority weren’t scheduled to return to session until Friday afternoon. Thats when hours of debate on budget bills could begin.

“This is the time of the budget process where this will blow up, this is closed, then this will blow up, then there’s last-minute things that nobody even talked about comes in, the last minute stuff people are trying ot put in,” said Senate Deputy Majority Leader John DeFrancicso (R-Syracuse) in an interview. He said the governor continues to sometimes raise the possibility of short-term “extenders” that would continue current levels of funding while negotiations continue on a late budget.

“I know things can come together very quickly,” said Sen. James Tedisco (R-Schenetady).