Mayor Marion Barry told the City Council yesterday that the District could have a budget deficit of as much as $60 million at the end of the current fiscal year if overspending by city agencies continues and if other costs -- such as pension and unemployment payments -- wind up as high as now projected.

Barry also warned the council members at a closed-door briefing that the city is facing a serious $60 million cash shortfall this summer, unless Congress moves quickly to authorize the city to borrow from private lenders. The bulk of that amount -- some $40 million -- is all due on April 1, when the city must pay back money it collected from a tax on professionals, which was later declared illegal by the court.

Administration officials had no comment on the briefing on how they plan to resolve the problems, that seemed to directly contradict Barry's assertion Wednesday that the current year's budget is balanced.

City Council members who attended the briefing took the news as a clear indication that the new shortfalls could swell the deficit of $388 million that had accumulated by last Oct. 1, the beginning of the current fiscal year. They believe they also could hamper efforts to secure borrowing authority from Congress and create more municipal financinal woes for a beleagured city government.

In an effort to avert a financial shortfall during the last fiscal year, Barry laid off about 1,000 city workers, held down paid increases for city employes and imposed a package of new and increased taxes and user fees. He also reduced some key city services, including cutting back on trash collection, and delayed a number of capital improvements projects.

Council member John A. Wilson (D-Ward 2), chairman of the council's finance and revenue committee, predicted yesterday that more layoffs and deeper service cuts would probably be necessary this year to offset the newly disclosed shortfalls.

In addition, Barry told the council members that he would have to get more money and spending authority from Congress to balance next year's budget, that already contains no funds for employe pay increases, in order to absorb President Reagan's proposed cuts in federally funded programs.

Ironically, the current projections that Barry said could throw this year's budget out of balance include $22 million in pension liabilities for city employes taking advantage of early out retirement provisions that the city sweetened in an effort to reduce the work force with fewer layoffs.

Sources attending the meeting said that apparently more city workers have chosen to leave the government than originally expected, and if the trend continues, the city would lack sufficient money in its pension fund.

Also, Barry said the city could have an unexpected debt for unemployment and disability insurance of up to $7.3 million. Neither the mayor nor any of the aides who attended the meeting with him would explain these unanticipated costs, according to council members who attended.

The most serious problem outlined at the meeting was overspending by city departments and agencies. Barry reportedly told the council members that the Department of Human Services could wind up spending up to $20 million more than was allotted in its 1981 budget. Much of that amount comes from Medicaid and costs for administering the Aid to Families With Dependent Children Program.

The Council members said Barry told them that the Department of Corrections could end up spending $7.7 million over budget, while the city's public housing agency could end up $4.1 million in the red because the rents being collected were not meeting the expenditures.

According to those who attended the briefing, Barry said that he did not want to "alarm anybody" by revealing the potential deficit, and he added that the potential cash shortfall this year may not be "significant in a municipal environment."

While Barry officials remained closed-mouthed about the city's financial situation, council members were swift to react.

Council Chairman Arrington Dixon said, "We're in big trouble," but later said that he hoped the disclosure would not hamper the city's effort to secure borrowing authority.