Sen. Patrick J. Leahy (D-Vt.), the chief congressional overseer of the city's budget, has told Mayor Marion Barry that the way he resolves the District of Columbia's budget crisis will be "a test of your administration."

In a stern letter to the mayor made available yesterday, Leahy, also said "the manner in which the budget problems are overcome should be the turning point of [District] budget autonomy and true home rule."

Leahy, chairman of the Senate District appropriation subcommittee, was prompted to write the letter by city officials' statements that there will be a $28 million deficit in the current budget, which runs through Sept. 30. City agencies have been asked to prepare budget cuts to overcome the deficit.

Other public officials have said the deficit could rise much higher and have asked Barry to explain how the figure was determined.

Leahy speculated in the letter that the deficit could run "anywhere from $30 million to nearly $200 million."

City Administrator Elijah Rogers said yesterday that the mayor was "concerned about the tone of the letter and concerned that the senator would involve himself with the city's budgetary problems at this time . . .We are on top of it, doing what we are supposed to do."

Barry smiled broadly at a meeting yesterday of his top officials when reporters asked him about the letter. He said only that he was preparing a reply.

Leahy also said in the letter, dated Jan. 31, that Congress expects to review any spending cuts imposed by the District of Columbia.

"It is clear," Leahy said, "that congressional approval is required for any significant departure from the programs" that were approved by Congress when it passed the city budget.

Leahy said that the City Council also should review any proposed cuts.

City budget director Gladys Mack said yesterday that the deficit projection and the city's way of dealing with it have a recent precedent. In January 1974, then-mayor Walter E. Washington imposed $16 million in budget cuts that halted all but the most vital purchases and the hiring of new personnel, she said.

Barry declined yesterday to give any hint of the type of cuts that department heads have listed as necessary if they must reach the $28 million cutback goal. The lists are being reviewed by budget officials before a recommendation is made to the mayor, possibly later this week.

In addition to a likely near-total hiring freeze, the potential cuts include a reduction of police patrols, trimming the hours of summer recreation programs, and possibly closing swimming pools one day a week and eliminating overtime pay for employes of institutions run by the Department of Human resources.

Robert Lewis, director of the city's licensing department, said he suggested that the work week for employes might be reduced and paychecks shrunk proportionately.

For such departments as labor and housing, which are financed chiefly by U.S. grants rather than by local taxes, the cuts would be minimal, officials said.

Although the school system has not submitted any list of possible cut-backs, budget director Mack said Barry is prepared to invoke powers granted by the city's Home Rule Charter to order a cut in school funding. The goal for school reductions is $8 million.

R. Calvin Lockridge, president of the school board, wrote Barry on Jan. 25 saying the board would defy the directive that it propose how it would absorb its share of the reductions. Barry said yesterday that he never received Lockridge's letter and declined to comment on it.

Leahy's House counterpart, Rep. Charles Wilson (D-Texas), has not contacted Barry and will wait for the city's proposal, an aide said yesterday. c