Mayor Bloomberg and the Democratic candidate for governor, Eliot Spitzer, are on a collision course over how the city and state will split the cost of settling the multibillion-dollar Campaign for Fiscal Equity schools financing case.

Mr. Spitzer, who is campaigning on a promise to swiftly resolve the 13-year-old case, is “looking for a major contribution from the city,” an informed source told The New York Sun.

The attorney general could demand that the city share as much as 40% of the total cost of complying with a court order requiring additional billions of dollars in operating aid for New York City public schools. Mr. Spitzer has said it would take $4 billion to $6 billion a year to satisfy the lawsuit, which is under appeal.

That puts Mr. Spitzer directly at odds with Mr. Bloomberg, who has said repeatedly that the state should be entirely responsible for coming up with the operating aid. The Bloomberg administration’s position has been that the state is shortchanging the city’s school system and should increase aid by at least $5.4 billion a year.

The question of how their differences will be resolved could be one of the first issues to test the relationship between Messrs. Spitzer and Bloomberg, who are on friendly terms in the public eye.

Sources close to the case expect both sides to soon begin the behind-the-scenes process of negotiating a compromise on the city’s share. Budget gaps are on the horizon for the city and state, which will only add to the pressure. New York State is facing a two-year budget gap projected to be between $9.8 billion and $13.9 billion. Mr. Bloomberg, meanwhile, has warned that the city would be forced to cut education programs if required to pick up a portion of the tab.

A spokesman for the mayor said the administration’s position regarding the lawsuit has not changed, but he would not comment on its differences with the Spitzer campaign. Officials for the Spitzer campaign did not return telephone calls for comment.

Mr. Spitzer has said the state could afford to pay for the first installment of the settlement under his $11 billion savings plan, which involves program consolidations, would increase the amount of money the state collects from bottle deposits, and calls for stricter enforcement of sales tax collections on Indian reservations.

For most of the history of the case, the dispute over who is responsible for paying the operating aid has largely remained in the background while the state and the plaintiffs have fought over how much is owed to the city school system.

In 1993, the Campaign for Fiscal Equity filed a lawsuit alleging that the state’s funding system shortchanged city students to a degree that violated the state constitution.The plaintiffs are demanding that city schools receive $5.6 billion in operating aid in addition to the funding levels for fiscal 2004-05.

The Pataki administration says it would take $1.93 billion a year to meet constitutional standards, but has submitted a plan to a panel of court-appointed referees that proposes a $4.7 billion increase. In a brief submitted to the Court of Appeals last month, the attorney general’s office said the $4.7 billion is a “policy preference, not a view of the constitutional mandate.”

Following the recommendation of the referees, a state Supreme Court judge in February 2005 ordered the state to increase operating aid by $5.6 billion. An appellate court last spring revised the ruling, saying the Legislature should consider a range between $4.7 billion and $5.6 billion.

The ruling was appealed by the Campaign for Fiscal Equity, which asked the Court of Appeals, the state’s highest court, to enforce Justice Leland DeGrasse’s order.

State courts have left it up to the city and the state to determine how the cost of the funding should be shared.

An opinion by the appellate division of the state Supreme Court, an intermediate court, states that “it is for the Governor and the Legislature to make the determination as to the constitutionally mandated amount of funding, including such considerations as how the funds shall be raised, how the additional expenditures will affect other necessary appropriations and the economic viability of the State, and how the funding shall be allocated between the State and the City.”

Mr. Pataki has said the city should pay 40% of the cost of the settlement because the city pays 40% of the school system’s budget, with the remainder covered by the state and federal government. The Campaign for Fiscal Equity has recommended that the city pay 25%.