The Prince William Board of Supervisors is sharpening its carving knife to pare at least $3 million from the county school board's $102.3 million budget proposal for fiscal year 1982.

School Board Chairman Gerard P. Cleary is taking the supervisors' promised budget cuts seriously, and said he would like to eliminate an additional $2 million -- for a total cutback of $5 million -- before sending a final budget proposal to the supervisors for approval.

The proposed budget is 13 percent higher than this year's budget of $90.7 million, and a majority of the supervisors recently told the school board to cut the increase to at least 10 percent.

With 35,674 students, Prince William County is the fourth largest school system in Virginia. It is also the only school district in Virginia with a dual schedule in which half the students follow a traditional nine-month academic calendar and the other half follow a year-round schedule. This is the last year for the plan, which had been instituted in an effort to alleviate overcrowding at several schools.

In his recent budget message, Superintendent Richard Johnson blamed inflation and rising personnel and energy costs for much of the budget increase. He also said he wants to expand the county's special education program and pay for 12th grade textbooks out of tax revenues during the first stage of a phased withdrawal from rental schoolbooks.

In addition, Johnson proposed giving all school personal a 9 percent cost-of-living increase that would cost more than $5 million.

"In these times of continuing inflation, the school system has an obligation to make sure that its employees keep pace with rising prices," Johnson said.

School Board Chairman Cleary, however, seemed more concerned with keeping a lid on large-scale budget increases. He targeted the proposed pay raises and the free textbook plan for the scrap pile.

"The 9 percent increase for personnel will be the first thing to go," predicted Cleary, who conceded the cutback might mean the county will have more problems with teachers. "Fairfax County is always breathing down our necks, hiring away our teachers, because they pay more. But the cuts have to come from somewhere."

Two other programs that Johnson is touting appear to be in jeopardy: a plan to hire more teachers to reduce the student-pupil ratio in grades one, two and three and a 100 percent increase in instructional supplies for language arts, math, science and social studies. Cleary said he expects both proposals to fall by the wayside in upcoming budget-cutting sessions.

School budgets comprise a combination of federal and state funds, with the local contribution coming from the county's general fund. For the coming year, the Prince William schools are asking for $48.7 million in local funds, a 26 percent increase over the country's current contribution of $38.6 million. Cleary said he wants to reduce the $10 million hike and lessen the burden on county taxpayers.

A major unknown, he noted, is what the state will contribute to the schools.

Last year, schools received $1,099 per pupil from the state. This year, the General Assembly has talked of increasing that figure to $1,184. "Until they make up their minds in Richmond, we won't know our final figures," said Cleary.

The House of Delegates and the Senate Finance Committee have approved a per pupil expenditure of $1,184. This full Senate is expected to vote on the proposal today.