The District of Columbia government, hard pressed to pay for basic city services, will be forced in a few months to dip into tax revenues for the first time to help prop up its deficit-ridden unemployment compensation fund.

The added financial burden was created by a new federal requirement that forces state governments and the District to pay interest for the first time on the billions of dollars they have borrowed collectively from the federal government to meet skyrocketing unemployment claims.

Under the provisions of the federal law, passed to discourage such borrowing, Mayor Marion Barry was forced to include $2.4 million in interest payments in the budget adopted last week by the City Council for the coming fiscal year.

Over the next three years, the city is expected to pay an additional $7.5 million in interest on the loans.

The deficit in the D.C. fund reflects a nationwide trend caused by the recession. This year the ranks of local governments borrowing from the U.S. Treasury have swollen to an all-time high--26 states plus Puerto Rico and the Virgin Islands. The borrowing averages about $1 billion a month nationwide.

Interest payments alone will amount to more than $400 million this year, according to the U.S. Labor Department.

"The system is out of balance," said one federal unemployment official. "The states are going to have to clean up their unemployment systems and the economy is going to have to get better or we can't continue to fund benefits as we are doing now."

On Tuesday, the City Council is expected to adopt a new unemployment compensation law proposed by Barry that will cut benefits to the unemployed and increase taxes on employers. Federal officials said the interest requirement, which applies to loans obtained after April 1982, was designed to encourage changes such as the District's new law.

"This the interest payment is a direct assessment against all D.C. taxpayers," said City Council member Charlene Drew Jarvis (D-Ward 4), head of the council committee that oversees the unemployment program.

Jarvis said the interest payment also forces District taxpayers to help subsidize unemployed people who live outside the city. Thirty-six percent of those receiving District unemployment benefits last year lived in Maryland or Virginia, Jarvis said. Laid-off workers receive benefits in the jurisdiction where they worked, not where they live.

Although $2.4 million is relatively small compared to the entire $1.9 billion city budget, it still is roughly equivalent to the $2 million that education advocates tried in vain to add to the appropriation for the public schools.

Before interest was charged on the loans, District tax revenues were not used in any way for the unemployment fund. Jobless benefits are financed by special taxes paid by the city's 18,500 employers. The District, all 50 state governments plus Puerto Rico and the Virgin Islands only administer the program, in conjunction with the federal government.

But when total benefits exceed employer contributions, the District and the other governments can borrow from the federal government.

Virginia, which in the past boasted of surpluses in its unemployment fund, will borrow for the first time this year. State officials said they expect to borrow $60 million, but plan to pay it off on time and avoid any interest.

Maryland, which has not had to borrow since 1977, may also borrow from the federal government this year, according to Lou Panos, press secretary to Gov. Harry Hughes.

At present, the District has one of the most liberal benefit programs in the country. The proposed changes in the city's law would cut the maximum number of weeks jobless workers can collect unemployment benefits from 34 to 26, bringing the District in line with most jurisdictions. Businesses would face a major increase in unemployment taxes, which combined with benefit reductions, is expected to eliminate a $60 million deficit in the city's fund by the end of 1986.

Even with the new law, the District is expected to borrow an additional $17 million this year. Without the changes, city officials said, the city would have to borrow nearly twice as much, $30.1 million.