D.C. School Board President R. Calvin Lockridge said yesterday that the school system may have to open three or four weeks late next fall because it lacks the money to open on time.

Speaking to about 95 delegates of the D.C. Congress of Parents and Teachers, Lockridge said the system expects to be about $12 million short by Sept. 30, the end of the fiscal year, and may have no choice but to delay school's opening.

He also said that virtually the entire administrative staff of 945 -- the largest of any school system in the area -- may have to be furloughed through the summer months and into September to make up for the $12 million deficit. Unlike teachers, school administrators normally work year-round.

Lockridge said the school board had been directed by the House subcommittee that oversees the District of Columbia budget to cut its administrative staff. That committee refused to approve $4 million special appropriation the school system said it needed to successfully cover its expenses for the current school year.

"We'll either have to furlough (the administrators) or fire half of them. And you can't fire half of them if you intend to open schools," Lockridge said.

Mayor Marion Barry's only comment on Lockridge's statement yesterday was that "school boards frequently blame mayors and city councils for their own ineptness," a spokesman said.

School board members Carol Schwartz and John E. Warren said it may be premature to talk about delaying the opening of school, Superintendent Vincent E. Reed said "I'll be looking at all the options. At this point in time, I can't make any definite statement," about the upcoming school year. Warren is chairman of the board's finance committee.

School officials anticipated a budget shortfall of about $24 million for the current school year and found ways to absorb about $20 million of that deficit without eliminating any school programs.

But Lockridge said the board does not yet know how it is going to absorb the additional $4 million, a $6.2 million cut Barry has ordered and a $2 million cut that Congress is expected to impose.

"I'm in favor of some budget cuts, but this is beyond reason," said board member Eugene Kinlow. "I don't know how we can be expected to absorb $30 million in cuts after the school year began."

Kinlow said "we may not have any other option" but to open schools later than usual. School traditionally opens right after Labor Day, and Lockridge said the delay might be until Oct. 1 or so.

Lockridge, who received two standing ovations after his speech to the parents and teachers, accused Barry of "treating the public schools as his personal bank account." Lockridge was alluding to a dispute between the mayor and the school system over a $20 million fund that the school system says it has saved up over the last few years.

City officials insist the money in the fund automatically reverted back to the city because the school system did not spend it in the years it was appropriated for.

Most of the money in that fund comes from federal grants or money that was in the general school budget in prior years.

City officials, Lockridge said, "are just looking and grasping for money [to ward off the city's anticipated deficit of about $172 million] anywhere they can find it and they are zeroing in on the school system."

The mayor has said his budget cutting requests to the school system are justified because the schools expect enrollment to drop to about 98,000 this year, 15,000 fewer than last year.

In addition to the cuts that must be made to the 1980 budget, the school board has yet to decide what to do about the budget for the 1980-81 school year.

Barry and the City Council have approved a budget that is $27 million less than what the system says it needs. And Reed has called for eliminating about 700 teaching positions, reducing the prekindergarten and adult education programs, increasing the pupil-teacher ratios in the elementary schools and closing several schools to absorb the deficit.