D.C. General Hospital, facing a potential $10 million deficit and a threat by its Board of Commissioners to close it down on July 1, will remain open with the help of at least $4 million in additional funds from the city, according to Gilbert Hahn Jr., chairman of the hospital's board.

Hahn told the City Council during hearings on the hospital's budget that he is still negotiating with Mayor Marion Barry over the remaining $6 million that the hospital says it needs to remain open through the fiscal year that ends Sept. 30.

But Hahn said major disagreements remain over the hospital's budget for the next few years. "I'm inclined to think right now that one way or another, we ought to get through the '83 year so that we can deal with '84 and '85," Hahn said after the hearings at the District Building. "Those are the real problems."

The city has proposed a fiscal 1984 budget of $43.1 million, about $10 million less than Hahn's commission requested.

Elizabeth Reveal, Barry's budget director, said the mayor wants the hospital to consider changing some of its operations to save money, including possibly eliminating its role as a teaching hospital, reducing administrative costs and increasing its occupancy rates.

Hospital officials have attributed rising costs to its low occupancy, reductions in Medicaid and Medicare payments and the practice of doctors of billing the hospital rather than patients for services.

In other budget testimony on the first day of detailed council hearings on Barry's proposed $1.99 billion budget for 1984, representatives of the D.C. Office on Aging and the Commission on the Arts and Humanities defended their increased budget requests, which come at a time when social services are being cut.

E. Veronica Pace, director of the Office on Aging, said its budget request of $5.1 million--an increase of $815,000 over what the agency is spending this year--was based on the mayor's commitment to programs for the elderly, which she said have increased "sevenfold" since 1979.

Mildred E. Bautista, head of the arts commission, said the increase of $504,000 in her budget request largely was to fund a grant program to institutions and individual artists that was rejected last year by Congress, which oversees the city budget.

"What would you have me say to people on the General Assistance Program," a social service function the city is eliminating, asked Council Chairman David A. Clarke, who also noted that the city is cutting the amount of funds it uses to pay utility bills for public housing tenants.

"Artists are people, too," Bautista responded, adding that many artists are poor and that "as times get more difficult economically," people depend on arts for motivation and entertainment relief from their problems.

"I think you have given an argument not to cut your budget," Clarke said, but added that the testimony also did not give any reason to increase the budget over current spending levels.