The city's financially beleaguered board of education managed to pay its 48,000 employees today after receiving a $37.2 million advance on its state aid.

Meanwhile, Mayor Jane M. Byrne announced a plan to bail out the school system, believed to be as much as $500 million in debt, by raising real estate taxes, selling $200 million in bonds to the state, and setting up a "fiscal oversight" agency.

"Everybody's rushed to the banks to cash their checks," a school board employe deadpanned. "They want to get their money before it runs out."

Byrne acknowledged that her plan has "a number of substantial legal and financial problems which must be overcome before it is implemented." There was no telling whether the board would get any new money in time for its next payday, four days before Christmas.

Banks are refusing to lend money to the school board, and Gov. James R. Thompson has said he won't approve a state loan until he is certain it will not jeopardize the state's triple-A credit rating. He and Treasurer Jerome A. Cosentino plan to discuss the matter in New York next week with bankers and officers of the bond-rating services.

"There's a difference of opinion as to whether the banks are wholeheartedly behind this plan," said a Thompson aide.

The plan also would not work unless the state legislature agreed to raise the school board's debt limit and to authorize the real estate tax increase. While suburban and downstate legislators have usually been happy to let Chicago tax itself, this time Thompson fears political repercussions.

The school crisis has embarrassed city political leaders, who used to take pride in contrasting Chicago's solid financial position with that of cities like New York and Cleveland.

Mayor Byrne has called the situation "a national disgrace," and accused business leaders and politicians of having "covered up" the true state of school finances for as long as 10 years. That would include the last seven years of the reign of the late mayor Richard J. Daley, whom Byrne and many others revere as a master administrator who made Chicago "the city that works."

The school system's financial records are said to be such a mess that independent auditors have been unable to determine the exact deficit after more than a week of study.

Byrne has accused School Superintendent Joseph P. Hammon of lying to her about the size of the deficit. Hammon, who at $82,500 a year was said to be the nation's highest-paid educator, was the first of four top school officials to resign since the crisis broke in mid-November. The others are school board president John D. Carey, who immediately left town, and the two top financial officers, Eugene J. Gutierrez and Robert Stickles.

The crisis hit Nov. 14 when Moodys Investors Services downgraded the school board's credit rating. This made it impossible to borrow $124.6 million via the short-term notes that the board sells to meet expenses while it awaits tax revenues.

Moody's said it had discovered that school officials intended to use some of that money to pay off other loans, a possible violation of state and federal law. The Securities and Exchange Commission began an investigation.

The board met its Nov. 21 payroll after the state advanced its December aid payment, but it still was unable to pay $44.2 million of bills due Nov. 30. It didn't turn over to the federal government $15.9 million in income taxes withheld from paychecks, another apparent violation of law. It also held on to almost $13 million in paycheck deductions for employes for pensions, savings, union dues and medical insurance.

And it missed a $15.8 million rent payment to Chicago's Public Building Commission, which builds schools and leases them to the school board. As a result, Moody's lowered the credit rating on some of the commission's bonds.

While the schools apparently have been mismanaged financially, it also is true that they have not had a real estate tax increase since 1971, even though the city's tax base has shrunk during that period.