Mayor Marion Barry, acknowledging that the District of Columbia will be at least $60 million in the red at the end of this fiscal year and financially squeezed for years to come, plans no new tax increases over the next two years but will make deep, long-term cuts in city services instead.

City officials and their financial advisers, in a series of recent interviews, described a daily juggling game to hold the deficit below the $170 million potential forecast in March, with remedies ranging from staff layoffs to a ban on the purchase of toilet-bowl brushes.

The city is not going to go bankrupt or miss any paychecks through the end of the fiscal year on Sept. 30, they said. But it will carry over some debts into next year and will be obliged to borrow more money from the U.S. Treasury to meet its current obligations.

Ivanhoe Donaldson, general assistant to the mayor and head of Barry's cash-management team, put the probable deficit at the end of the current fiscal year at $60 million to $80 million.

Other analysts of the District's finances, however, said that the $170 million figure was never valid and that the probable real deficit will be over $100 million. A similar deficit is possible next year.

Either way, the city's accumulated budget deficit, estimated by private auditors at $284 million at the close of the 1979 fiscal year, is likely to grow.

Faced with increasing pressure to bring income and spending into balance, the Barry administration has decided that payrolls and services will have to be slashed in order to hold the line on taxes.

The clear message to all who live and work in the city is that years of austerity lie ahead; the cutbacks and layoffs already announced are just the beginning.

The process will be traumatic, a shirt-sleeved Barry told news executives at a briefing in his office last week, but the reductions are unavoidable because, "We are cutting out our credit cards for the future."

Donaldson said the city will have to live within its income, which means, "If we can't afford it, we aren't going to do it."

Donaldson said the District is already "taxed to the limit" and that further increases might be counter-productive, driving out businesses and middle-class residents -- a view shared by John A. Wilson (D-Ward 2), chairman of the City Council's finance and revenue committee.

Most key city taxes -- including sales, gasoline and commercial property taxes -- have already been rasied during Barry's first 18 months in office, and many are among the highest in the area.

An exception is the residential property tax. But any attempt to raise that would encounter strong City Council opposition and would be politically risky if Barry sought reelection in 1982.

If Barry can avoid asking for further new taxes on District residents and businesses between now and the fall of 1982, it would be a remarkable fiscal and political achievement, brought off at the cost of what he called "substantial" cutbacks in services."

The city has already borrowed $60 million from the U.S. Treasury to remain solvent this year and has asked the federal Office of Management and Budget for authority to borrow more.

Carter administration officials say that request will be granted, which in effect gives the city an open line of credit to meet payrolls and keep essential services functioning.

Throughout what has been a very tough year for Barry's fledgling administration, one problem has been getting a firm handle on the city's financial situation.

The $38 million computerized accounting system designed to enable the city to manage its finances is, in Donaldson's words, "nonusable from a management standpoint."

He, the mayor and other highranking officials have computer terminals in their offices, but most transactions have to be confirmed by hand. It will probably be January, when an audit is to be completed, before a complete assessment of the city's fiscal performance this year will be available.

Neither Barry nor his aides will say how much they believe they have saved through cutbacks, layoffs and a hiring freeze already imposed. But some city officials are predicting that the city will not only spend more than it takes in revenue, but because of an inability to closely monitor and control some departmental outlays, it will also spend more than it was permitted to by Congress. That would be a technical violation of the law by the image-conscious Barry administration in its first full year of budgetary responsibility.

The consequences of exceeding the congressional limit are unclear, but they do not include the shutdown of the city government. Since none of the participants in the process wants to provoke a crisis -- such as a payless payday for city workers or a delay in the opening of schools -- there is not likely to be any, sources in the city administration, Congress and the federal government agree.

But that does not mean the budget crisis is illusory. The city is spending more money than it has adding to the cumulative deficit officially calculated by the city's auditors, Arthur Andersen & Co. at $284 million.

The search for a formula that will enable the city to repay its loans, meet current obligations, eliminate the deficit and spread the burden equitably among its citizens is consuming the time and energy of Barry and his staff.

The desire to avoid raising taxes is one of the few clear-cut pieces of a complex puzzle -- a puzzle made more complex by the fact that no one person or body of officials has full control of it.

Another clearcut piece is the mayor's belief that without new taxes, there is no choice but to slash payrolls and services in a process that is going to have to continue for the foreseeable future.

Barry has promised to deliver a comprehensive plan for solving the puzzle by July 21. Few details have leaked out, but a briefing for news executives last week he offhandedly said that he did not envision asking for new tasks through the 1982 fiscal year.

He told a congressional subcomittee that "hard, excruciating, painful and difficult decisions" have to be made about what services the city will continue to provide and how to pay for them.

"It's like getting to the supermarket checkout counter with $140 in groceries and you only have $120," he said. "Your children are tugging at you, but some of those things have to go back on the shelf."

There will be no "quick-fix solutions," he said, and whatever long-range program he develops for bringing the city's finances under control will require a "deal" with the federal government.

The mayor and his aides are scheduled to brief White House officials in a few days on some of the proposals they are considering in an effort to obtain Carter administration support for a package that will surely require action by Congress.

Meanwhile, just staying afloat is a daily concern at the District Building.

The mayor receives a daily report on the city's cash balance. He has declined to reveal what these reports show, but the squeeze is so tight that Donaldson and other top members of the staff spend hours every week reviewing individual purchase orders and vouchers, down to the level of a $2.02 supplementary Pepco bill that Donaldson rejected last month.

"It cost more than the amount of the bill to process the paperwork," he said. "They should have just left it to be added to the amount of the next regular monthly bill."

He said he also vetoed a request from the Department of Recreation to buy 47 toilet-bowl brushes for restrooms at swimming pools because the department did not explain what happened to brushes purchased last year.

"We are going to have to live within our incomes," Donaldson said. That means that "the process of cutting programs and services is going to have to continue. But how far can you go when 24 percent of the people in this community are dependent on the District government in some way for [social] services?"

He said service reductions were contrary to Barry's entire philosophy. "This mayor came into office to do more not less," he said. "But what is the extent of our responsibility? Safety and water, yes, but what are we going to do beyond that?"

Donaldson said that the city "can't control the transit subsidy or Medicaid costs, so how much subsidy can we afford for our university? It was $48 million this year. Maybe some of the students will have to help out."

The mayor himself went on television last week to stress that the city lack the resources to meet the needs of its citizens, saying that such problems as housing and care of the elderly would have to be dealt with by the federal government.

There is one school of thought, among some city officials and outside observers, that the mayor is deliberately emphasizing the discouraging news about the city so he can pick up political points if things turn out not to be so bad.

But even if that is ture, the consensus among business leaders, congressional staff aides and other experts is that the situation is grim enough without embellishment.

"Look at the budget for fiscal '81," which starts Oct. 1, said city auditor Matthew S. Watson. "It shows a cash reserve of only about $700,000 at the end of the year. That's about an hour-and-half payroll. Even if the revenue projections are only low by 1 percent, they would be short by $9 million. There's no margin for error. A single snowstorm could wipe us out."

That 1981 budget, now before Congress, is theoretically balanced, as the law requires, with proposed expenditures and revenues of just over $1.5 billion. But is presumes several things that are not likely to happen.

It presumes that Congress will permit the city to lay off 204 police officers, will give the District the full authorized federal payment of $300 million, and will permit the city to levy property taxes on pieces of prime real estate that are now exempt, such as the midtown headquarters of the National Geographric Society. It also presumes that Washngton's police officers and firefighters will accept less than the 9 percent pay increases won by their New York counterparts.

If congress obliges the city to retain 204 police officers Barry wants to lay off and rejects the additional revenue sources, further service cuts will have to be made in other departments. Wilson, chairman of the City Council's committee on finance and revenue, predicted that that is exactly, what will happen.

"The '81 budget is by my estimation out of balance by $99 million," Wilson said in a telephone interview. Even if Congress grants the mayor's requests, he said, "We will be short by $14 million to $40 million."

Wilson predicted that some time early next year the mayor would be obliged to ask for tax increases, but, he said, "There's no place left to tax. There can be no more taxes on Business. All that's left is the individual property tax rate, and I just won't tolerate that. I'm totally opposed to it and it will never leave my committee."

Since "there is no tax that can be increased at this point," Wilson said, "what has to happen is a total reorganization of the government. They are shifting some people around, but I am talking about total reorganization.

"Whether they (Barry and his aides) are considering it or not, it's the only solution, and I can get seven votes for it."