The Alexandria School Board last night approved a $59.9 million budget request for the next school year, fanning a dispute with city officials over how much money the city's schools need to operate.

City Manager Vola Lawson, who is expected to make public her proposed city budget in the next few days, had asked the board to send her a request seeking $1.5 million less, or a 6 percent increase, instead of the 7.9 percent increase requested unanimously last night.

The City Council will ultimately decide the matter when it votes on the final city budget in May.

School Board Chairman Lou Cook said that if the schools do not get the extra $1.5 million, class size will increase and the number of teachers will be cut.

"If we are serious about remaining competitive with Arlington and Fairfax, and in a few years with Loudoun and Prince William," said School Superintendent Robert W. Peebles, "we must approve the increase."

The budget request includes teacher salary raises averaging 7.7 percent, and would add $233,715 for minority achievement progams, $222,211 for special education programs and $216,475 for a new transitional level between kindergarten and first grade.

Teachers' salaries currently begin at $18,200. The average salary for the system's 751 teachers is $34,000.

The School Board pledged to bolster minority achievement programs after scores on 1984-85 standardized tests showed startling gaps between black and white students.

"You get what you pay for," said board member Lynnwood G. Campbell Jr. Without the new funding, he said, "minority achievement will become minor achievement."

On a 100-point scale, 11th grade black students scored 48 points below their white classmates last year on the Science Research Associates test.

The special education programs serve 1,333 of the system's 10,344 students, including the handicapped and those enrolled in English as a second language programs.

Formation of a transitional primary grade as a pilot program is designed to help youngsters who do not have the skills or who are not mature enough for the first grade.

Referring to problems of funding the school budget, City Council member Robert Calhoun, who attended the meeting, said, "There's a large gap and we're going to have problems closing it." Earlier in the day, Mayor James P. Moran Jr. said the council would not be able to approve anything more than a 6 percent school budget increase.