ALBANY — Schools are increasingly leery of asking for voter approval to override the state’s property-tax cap, but they said they are managing by making cuts to programs and draining reserve accounts.

A review of state records by Gannett’s Albany Bureau shows that the 669 districts in New York are proposing to use about $262 million more in reserves than last year, up about 18 percent, to balance budgets for the 2013-14 school year, which starts July 1.

“Once school districts exhaust these fund balances, there will be nothing standing between them and fiscal or educational insolvency,” warned Michael Borges, executive director of the state Association of School Business Officials, in a statement Wednesday.

Only 28 school districts plan to seek an override of the property-tax cap when residents vote May 21 on the school budgets. That means 96 percent of schools plan to stay within the tax-cap limit, which on average would allow them to raise taxes by up to 5 percent.

Schools instead are seeking an average tax-levy increase of 2.9 percent, records from the state Education Department showed. With a 4 percent average state aid increase this year, schools were able to use reserves to limit tax increases, school officials said.

“This year’s state aid increase has helped avert some layoffs and program cuts and, in some cases, has allowed districts to restore positions and programs,” said Timothy Kremer, executive director of the state School Boards Association.

The tax cap, implemented last year, limits the growth in property taxes to 2 percent a year or the rate of inflation, whichever is lower. But there are exemptions to the cap — such as major growth in pension costs — that leads to a different levy limit for each school district.

Gov. Andrew Cuomo has touted the success of the tax cap, which applies to schools and local governments, as a way to curb among the highest property taxes in the nation. In 2011, the median property tax paid by a homeowner in New York was $4,090 — twice the national median.

“This shows the continued success of Gov. Cuomo’s tax cap,” said Cuomo spokesman Richard Azzopardi in a statement.

Only 23 percent of local governments this year planned to override the cap. And there are fewer schools seeking an override this year. Last year, about 50 did.

Overriding the tax cap for schools is not easy. It requires approval from 60 percent of voters. On the first budget vote last year, 24 budgets were rejected — and 19 of those were in districts that sought an override.

The penalty is steep for a district that fails twice to pass a budget: They can’t increase taxes at all for the upcoming fiscal year. For districts that fail to pass a budget May 21, a re-vote would be held June 18.

Statewide, districts are proposing to use nearly $84 million in reserves to limit tax increases, dropping their total account balance by 6 percent to nearly $1.3 billion.

Twelve of the districts seeking an override are from Long Island and Westchester County. But those schools get less state aid, and therefore rely more heavily on property taxes to cover their costs.

Four districts are seeking an override in Westchester: Scarsdale, Briarcliff Manor, Ardsley and Irvington.

Briarcliff Manor has drained its fund balance, leaving just $125,000, state records show.

The district is seeking a tax-levy increase of 4.85 percent, though its levy limit is 3.18 percent.

“Our budget has decreased over the last five years,” said Stuart Mattey, assistant superintendent for finance. “But we’ve been using fund balance to sort of offset any increase in taxes over that time period, and you can only go so long doing that.”