The fortunes of Chicago's public school system had sunk so low by the weekend that Major Jane Byrne suggested Saturday that a White House conspiracy was to blame.

She called the Carter adminstration "oppressive and stupid" after Justice Department lawyers asked Attorney General Benjamin R. Civiletti to sue the Chicago Board of Education unless it committed itself within 30 days to an acceptable plan to desegregate city schools.

The mayor's anger had a new target late Friday when school board officials told her the Internal Revenue Service was demanding payment of almost $16 million in withholding taxes owed the federal government before any of the school system suppliers were reimbursed for delivery of goods and services already delivered.

The school board is in such a financial bind that it failed to pay the $25.9 million in state and federal taxes and voluntary deductions it withheld from November teacher paychecks.

New school board President Catherine Rohter said that the schools might have to be shut down next week before the scheduled Christmas vacation because bus drivers, milk delivery firms and other systems vendors haven't been paid. Robter said she could not promise to meet the $41.5 million payroll that falls due this coming Friday.

Though accountants have worked around the clock for more than a week grappling with the school system's books in search of some possible out, the experts still don't know how much cash the board has. They estimate, however, that the schools need about $630 million to meet short-term and long-term obligations for the next two years.

The crisis began almost a month ago when leading bond rating houses gave the school board's credit risk a lower valuation after learning that the board, a taxing entity independent of the city, was using bond income to meet operating expenses. The board's ratings dropped to the point where no one would purchase the bonds on the financial market.

Salary checks for teachers were distributed a week ago only after the state treasurer agreed to an advance of funds normally due the schools in January.

But a bail-out plan conceived by the mayor, which called for the state of Illinois to make an initial $200 million loan to the schools over a three-month period, was later termed "unacceptable" by Republican Gov. James Thompson.

Meanwhile, Downstate leaders of the state legislature, which would have to approve a property tax increase, a rise in the board's debt ceiling and the issuance of several million dollars in long-term bonds under the Byrne plan, said they were reluctant to help the schools.

Left without so much as a plan to prop up the schools. Byrne was infuriated by the IRS demand that it be paid before anyone else with claims on the board. Suggesting the action might be official retribution for her much-publicized midstream switch of support from President Carter to the presidential candidacy of Sen. Edward M. Kennedy, Byrne, called the IRS action a "disgrace."

"I believe it is the right of every citizen to work out a payment plan" with the IRS when fiscal woes loom. Bryne told reporters today while delivering Christmas trees to residents of a low-income housing project on the City's South Side.

"I think [the IRS move] is political and I think they ought to come out from the White House, come out here and tell these mothers that the children aren't going to school next week because the federal government needs the money and they can't wait two weeks."

Don Williams, chief of the collections division in the Chicago IRS office, denied that politics was a factor in the agency's directive to the school board.

But Byrne said she was hoping to call the Illinois congressional delegation home from Washington to discuss means for bringing pressure to bear on the IRS.