WARWICK — The School Department administration last night presented the School Committee a $169.4-million budget proposal that leaves unanswered the question of how the schools can operate next year without running millions of dollars in the red.

The budget proposed by Supt. Peter P. Horoschak for the year that begins July 1 is up about $11.6 million, or 7.35 percent, from this year’s budget of abut $157.7 million and would fall short on the revenue side by about $4.7 million.

That projected deficit is based on the assumption that the city will give the schools the maximum local allocation it can under a 5 percent state tax levy cap and that state aid will be no less than it was this year.

If either of those assumptions proves over-optimistic, the deficit would increase.

“The limit placed on increasing the local school appropriation –– 5 percent — and the prospect of no increase in state aid to education have created a severe gap between School Department obligations and its ability to adequately fund the requisite programs and operations,” Horoschak said. “This results in the presentation of an unbalanced budget.”

School officials have known for months that the district would be facing a potential deficit of $6 million or more next year due to a variety of factors –– the most costly being the need to comply with its contractual obligation and pay the district’s teachers about $4.78 million in back pay come September.

In an effort to winnow down the projected deficit, the School Committee made the unpopular decision in March to close three elementary schools for a projected savings of $2.7 million. Two of the schools — Rhodes and Potowomut — will be left empty, while Drum Rock will be converted into an early-childhood center to house pre-kindergarten and other programs.

The big question that remained after the decision to close the schools was where else the district would be able to find savings.

The superintendent’s proposal keeps current educational programs intact but cuts out some nonteaching personnel including clerks, custodians and at least two teacher’s assistants.

Horoschak said there is little latitude in finding savings because the schools’ contract with the roughly 1,000-member teachers union limits to 20 the number that can be laid off in any one year. The closing of the elementary schools ate up that number, he said, and in fact, the district is left with nine more elementary teachers than it needs.

He said he hopes those positions can be eliminated through retirements.

School officials were still giving details of the proposed budget about three hours into last night’s meeting, but School Committee Chairman Christopher E. Friel said in an interview that the presentation of a budget with a deficit is “unacceptable.”

“It is disheartening that a projected deficit of this magnitude still exists despite the cuts that have already been made and the consolidation of the elementary schools,” Friel said.

“The School Committee will utilize all reasonable efforts and attempt to further bridge this projected deficit and fulfill our legal obligation to present a balanced budget.

“The School Committee is somewhat limited in the areas in which we can realize additional savings due to certain contractual constraints, for example, the limit of teachers subject to termination pursuant to the Warwick Teachers Union collective-bargaining agreement.

“[Still] we will continue to study all areas where savings may be realized.”

The School Committee will meet again tonight on the budget, with a session scheduled for 6 p.m. at Winman Junior High School, and it is expected to adopt a proposed budget at a meeting next week.

The board’s budget request goes to the mayor, who can change only the bottom line. After that, the mayor’s budget, including spending for the schools, is submitted to the City Council.

Once a final figure is set by the council, state law dictates that the School Committee adjust its budget accordingly so it has a balanced spending plan for the coming year.

Although teachers’ salaries have historically accounted for roughly 89 to 93 percent of the budget, Sam Holtzman, president of the Warwick Teachers Union, said their contract and the limit on layoffs is not the reason for the deficit. Noting that $4.7 million is a small percent of the total budget, Holtzman suggested that the school district look elsewhere, possibly to administrative positions, for savings.

bpoliche@projo.com