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It’s still the most expensive high school ever built in Massachusetts. But by the time the last blade of grass sprouts on the athletic fields this spring, Newton North High School will likely be finished slightly under budget.

As Newton officials close out the accounts for all items that went into building the new school, they are discovering that they shaved some costs.

Bob Rooney, Newton’s chief operating officer, said staff members are still finalizing the numbers, but he expects the project will be at least $4 million under budget, and perhaps $5 million less than the anticipated $197.5 million that officials had set as the maximum spending figure.

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“I think we’ve done a remarkable job,’’ Rooney said.

The savings are being tallied despite the last-minute discovery last year that the old high school building’s asbestos problem was much more extensive than anticipated. Newton had budgeted $7.2 million for the demolition of the old Newton North, but when workers discovered asbestos in the exterior walls, forcing them to dismantle the building brick by brick, the cost almost doubled to $14 million, Rooney said.

Much of the savings is based on the city not having to drain the project’s contingency budget to finish the school, Rooney said.

Construction costs were tightly managed, especially after all the concern over the school’s cost, said Alderman Lenny Gentile, chairman of the board’s Finance Committee.

The Newton North High project became the symbol of school construction excesses.

The project was conceived in 2000 as a $40 million renovation, but by 2006 had evolved into a replacement school with an estimated cost of $141 million.

The price tag kept rising, however, due to construction costs and unforeseen changes and delays, prompting some aldermen and other critics to call for a halt to the plan.

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The city’s longtime mayor, David Cohen, who oversaw the project from the start, decided in 2008 against running for reelection, and was out of office by the time the new high school opened in September 2010.

Jeff Seideman, cochairman of Newton for Fiscal Responsibility and a former candidate for alderman, said the lower costs are minimal, and that the city should have renovated the old high school instead of building such an expensive new facility.

“It’s still outrageously over the top,’’ Seideman said.

Because the city is paying the debt service on the high school’s construction loan - between $7 million and $9.5 million annually over the next 25 years - out of its operating budget, it is difficult to address other infrastructure needs, Seideman said.

Still, the reduced costs for Newton North should free up some money for other building needs, Rooney said.

There’s still uncertainty about how much below budget the project will be, with some final touches on the school still to be completed.

Crews need to put up foul-ball netting to keep baseballs from crashing into neighboring homes and complete the landscaping, Rooney said.

But Newton officials say they are certain that they won’t have to borrow another $4 million on an $11 million loan offered to the city by the Massachusetts School Building Authority, Gentile said.

“I expect the numbers to be better,’’ he said.

City staff will provide aldermen with a more detailed report on the costs next month, Rooney said.

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Deirdre Fernandes can be reached at deirdre.fernandes@ globe.com.