THE DEEPEST and most controversial cut in the mayor's proposed city budget for fiscal year 1981 is in the money the city would allocate for the public schools. In August the school board asked for an increase of about $32 million over its current budget. But the mayor's budget proposal for the school system would give the schools about $10 million less than they have to work with now.

Much of the difference of opinion between the board and the mayor on how much money the school system needs to operate is steeped in politics. Mayor Barry has said he thinks the school board is cowardly in not standing up to public pressure and closing city schools that have too few students. By limiting the board's budget, the mayor is trying to force it to do what he thinks is right: close some schools. Barry administration officials say the budget cut won't hurt the schools if services -- from teachers to books to heating oil for buildings -- are consolidated in a smaller number of schools operating near full student capacity while schools with too few students are closed.

The school board has told the mayor that it is an independent public agency -- not accountable to the mayor -- and it will decide when and where schools should be closed. It cites the danger of students' crossing major avenues as they walk to schools far out of their neighborhoods as one reason for its reluctance to close some sparsely populated schools last year. Board President Minnie S. Woodson views the mayor's reason for cutting the school budget -- declining enrollment -- as simplistic. She says fewer students won't mean that the school system will not have to pay an additional $10 million in annual raises to its employees next year. And fewer students, she has said, will not mean that the cost of the food fed to thousands of students daily will be less or that the price of books, chalk and chairs will be lower. The bottom line in Mayor Barry's budget, says Mrs. Woodson, is larger classes in the city schools and less progress toward improving the city's troubled school system.

But Mayor Barry and Mrs. Woodson may not be all that far apart in the way they see the school system. Mrs. Woodson may be right in saying that the mayor's focus on students when setting a new budget is simplistic, but the mayor's budget approach highlights what may be the school system's biggest problem: it is performing too many social services -- and paying too much for them -- when it should be intent on delivering education. For example, the school system feeds children; high schools have job centers; some schools even have day-care centers for their students who have children; and counselors are as accustomed to dealing with neglected or disturbed children as they are to handling college applications.

The solution to the budget conflict between the school board and the mayor may be for the mayor's city agencies to take over many or most of the social services now operated by the school system. That would leave the school board to work on making our city schools first-rate. To Mr. Barry's satisfaction, the school board could provide better education for a smaller price since it would not have to bog down in the social services. And the social services would probably cost less when operated within departments of the city government that already do this type of work. A side benefit -- and an important one -- to this kind of hand-in-hand cooperation would be to allow school closings to be discussed in terms of better education instead of the saving of money.