Mayor Marion Barry, firing heavy verbal salvos at the D.C. Board of Education, is fiercely resisting a move in the City Council to add $10 million to the public school budget for next year.

He said in an interview that he is "going to fight" the proposal, which was circulated last week by Council Member Betty Ann Kane (D-at large). He called her plan "ridiculous," and said she is "playing silly games" that would "mortage the future of the city" by giving more money to a school board that "refuses to tell the people how they're spending their money."

The District's public schools have suffered the heaviest losses in Barry's attempts to halt the city's deficit spending and to balance his budget by cutting payrolls and services. Over the past two months, relations between Barry -- a former school board president -- and the current board have deteriorated from cool to hostile, with the mayor accusing the board of refusing to submit justifications for its budget request and school board president R. Calvin Lockridge accusing Barry of "a systematic vendetta against the public schools."

With its $265-million budget request for the fiscal year beginning next Oct. 1 slashed to $238 million by the mayor, the school board has taken its appeal for "full funding" to the City Council. Members of the Council, sporting the same "full funding" buttons worn by the angry parents who have crowded their budget hearings, have been poring over the mayor's city-wide budget looking for funds that could be transferred to the public schools, creating a council-school-parents alliance that has isolated Barry in his determination to hold down school spending.

The search for funds that could be transferred from other departments to the schools produced only small amounts until Kane pounced last week on a little-noticed item in the city budget -- the debt service fund. That is the money set aside each year to repay loans from previous years and the interest on them.

Barry's proposed budget for the 1982 fiscal year requests $166.7 milllion for debt service -- an increase of $34 million over this year's amount. A good part of the increase is attributable to a change in repayment procedures suggested by the city's independent auditors, and that change is at the heart of the disagreement between Kane and Barry. The money owed to the U.S. Treasury, the District's only source of loans. The Treasury allows the city to wait two years before beginning repayment of any long-term loans, thus deferring the interest payout, but the auditors have recommended that the city forego that privilege and begin paying back its new loans immediately. That, the auditors persuaded Barry, would better prepare the city for the changeover to borrowing on the commercial market, which is the city's objective.

But Kane, also a former school board member, is arguing that there is no point, in a time of fiscal austerity, in accelerating the repayments when the Treasury is not asking for that. By sticking to the old system and reducing the amount Barry actually proposes to borrow, Kane told her colleagues the city cold cut the debt-service increase by about $10 million and transfer that money to the schools.

"The budget calls for increased borrowing and the accelerated repayment," Kane said. "But you can't change two policies at the same time when both changes are going to cost you money." She said she would move formally at a council meeting next Monday to make the transfer of the money to the school budget, and that several of her council colleagues have "expressed great interest" in the proposal.

But Barry said in an interview that he opposes Kane's plan for two reasons: He wants to accept the recommendation of the auditors and he does not want to give any more money to the schools in a period of declining enrollment when he is not satisifed that the school system is being operated eficiently.

"I think it's an irresponsible act," he said of Kane's maneuver, because "our accountants have recommended that we get ourselves on the right foot for once in life and have the money in the bank even though it's not due till a year later. If you want to go into the private bond market [which the city does, because interest rates there are lower than treasury rates], you can't play these silly games. . . . I think it's time for responsible, elected officials to stop playing those kinds of games and bite these tough bullets like I've been doing and put the money aside."

He said that if Kane wants more money for the schools, "she ought either to advocate a tax increase, which she won't do, or she ought to advocate a reduction in somebody else's budget, which she won't do."

Beyond the question of how to get more money for the school system, however, if the question of whether the school system deserves it. Barry's position is that the school board refused to submit detail schedules showing how it spends the money it receives. The mayor's "executive budget" for 1982 states flatly that "the Board of Education did not submit a budget to the executive branch" for 1982. "Despite repeated efforts to secure useful progammatic data on which to base a [spending] mark" for the school system, "the Board of Education did not provide a meaningful response."

The mayor cannot tell the Board of Education how to spend its money, but he can, subject to City Council approval, determine how much money the board should get. His budget director, Gladys Mack, assigned to the public schools a spending figure for 1982 that is actually $6 million less than they had this year -- a year in which budget cuts forced the schools to lay off about 1,000 employes. Mack arrived at her figure by a simple arithmetical formula based on estimates that enrollment will decline by 6,600 students next year.

Lockridge responded with public charges that the mayor is a liar who is trying to undermine the school system in an effort to control it. He published a scathing 33-page report saying that "the Barry administration appears hellbent on destroying the D.C. public school system." He said that one of Barry's letters to him about the board's budgetary procedures "was unique: the distortion of fact and lies contained therein reached new zeniths even for the Barry administration." Lockridge said that the board's budget presentation this year was the same as it has always been, and that the mayor know that detailed accounts of the schools' operations are always published in January.

But Barry seems determined. "If you ask the chairman of the City Council and the members how the schools are proposing to spend their requested $265 million," he said, "they can't tell you. They can't tell you because the school board will not tell the public what they're spending their money on. If you ask them how many textbooks were purchased in '80 and how much was spent, they won't tell you."

He said it would be "irresponsible to just willy-nilly give carte blanche to the $265 million; it's like saying to the public, 'I'm going to recommend a $50-million tax increase,' and someone is going to say, 'What are you going to do with it?' and I'm saying, 'I'm not going to tell you, I'm just going to get it.'"

The mayor complained that when the school system was forced to lay off workers "they went right to the teachers, right to the classroom" rather than reducing administrative or maintenance staffs. According to Barry, about 700 of the 863 persons laid off so far this year were either teachers or educational aides who worked in the classrooms. "They didn't go to any administrators, or if they did it was just two or three, and I think that's just not the way to do business. It's outrageous."

The mayor said that he had visited several schools and found examples of waste and inefficiency that made him question whether the system is well run.

"I've got inefficiency in my administration, too, no question about it," he said, "but at least I know where some of it is and try to correct it."

Barry said he knew his stand on the school budget might be politically unpopular, but concluded: "I'm prepared to take my lumps on this."