The Chicago Public Schools actually need $596 million to keep schools open the rest of the school year, and not just the $129 million officials have publicly discussed, a top aide to Mayor Rahm Emanuel said Tuesday.

The larger figure takes into account delays in receiving block grants from the state, according to Carole Brown, Emanuel’s chief financial officer.

Brown said all options are on the table to find the cash needed to stave off an early closing of schools that CPS CEO Forrest Claypool has warned might be needed but that Emanuel has ruled out to preserve the longer school year that cost the mayor a teachers strike to achieve.

Brown wouldn’t even rule out another tax increase — in addition to the $250 million already imposed for teacher pensions and the $45 million for school construction.

But she said that would not solve the need for an immediate cash infusion.

“It’s a really difficult problem we’re trying to solve and a difficult needle we’re trying to thread because ultimately the responsibility of funding CPS is the state’s, it’s not the city’s,” Brown said.

Other possibilities include: taking a “bridge” loan from tax-increment financing districts that might never be repaid; another round of borrowing; more cuts of school support staff; delayed payments to CPS vendors; and a request to the Chicago Teachers Pension Fund to delay part of the $721 million payment that’s due June 30.

A 60-day delay on the pension payment would solve some of the cash-flow issues since tax revenue typically rolls in by August.

Brown said “any solution that the city supports and that CPS puts forth will be a solution that does not put city finances in peril.”

A CPS spokeswoman did not respond to messages seeking comment on Tuesday. CTPF head Chuck Burbridge said the district will ask that “the $250 million that is to be received in July and August from the dedicated property tax levy will be applied to the June 30 payment,” using a policy that includes tax money that rolls in within 60 days of the fiscal year’s end.

Top mayoral aides had been scheduled to brief aldermen behind closed doors Tuesday on Emanuel’s plan to provide a temporary financial rescue for CPS. But the briefings were postponed for a second straight week.

Brown said the holdup is that the problem is much bigger than the $215 million in state pension help that Gov. Bruce Rauner vetoed — money that Claypool had counted on in CPS’ budget.

“We’re not ready because it’s a very complex, difficult problem,” Brown said. “They’ve got about $129 million left of the $215 million that they need to identify alternatives for. They also are dealing with a delay in block grants . . . of approximately $467 million. That is also an issue because they had budgeted that they would receive their fiscal year 2017 block grant in 2017.”

She continued, “The solution that we’re looking at may or may not include any funding help from the city. Everything is on the table.”

Rauner spokeswoman Eleni Demertzis blamed Democratic Comptroller Susana Mendoza, “Any outstanding payments from the state lie with Comptroller Mendoza, and can be processed when her office chooses.” She also said CPS received more state aid this year than last, despite declining enrollment, fewer students considered poor and rising property values.

Mendoza spokesman Abdon Pallasch said absent a state budget, his office lacks the cash to make more payments.

“Like the check bouncer who yells at his bank for bouncing a check from an account he himself emptied, the Governor disingenuously blames the Comptroller for not writing checks from state coffers that Governor Rauner emptied by failing his constitutional duty to propose a balanced budget,” Pallasch said.

CPS is in a bind for a second straight year after balancing its budget by counting on state money that had strings attached or didn’t come in on time because of Springfield’s budget standoff.

After Rauner vetoed the teacher pension bill, Claypool ordered more mid-year budget cuts and filed a lawsuit alleging that state school funding discriminates against CPS’ mostly poor and minority students.

The same day that a Cook County judge dismissed the lawsuit, Emanuel said he’d find a way to keep schools open.

The Sun-Times reported Monday that the mayor opposed Claypool’s lawsuit and didn’t know of his CEO’s threats to cut 13 school days until after they’d been made public, increasing the tensions between the two men.

“The school year ends June 21,” Brown said. “The fiscal year ends June 30. They will be in budget discussions for fiscal year ’18 in June. So our hope is literally, in the next few days, we will have something to present to the aldermen — because we’re gonna need to.”