The city comptroller, William Thompson Jr., a potential 2009 mayoral hopeful, may be attempting to turn himself into the education candidate.

In the latest of a series of attacks on the education legacy of Mayor Bloomberg, who has promoted himself as the education mayor, Mr. Thompson yesterday criticized Chancellor Joel Klein, saying he has mismanaged the city school system’s finances.

“Accountability begins at the top,” Mr. Thompson said at a Fordham University conference for city principals on fiscal accountability in education. Mr. Klein sat facing him a few feet away.

Calling Mr. Klein a friend, he said that nevertheless, “We have concerns.”

“The use of noncompetitive bids has soared out of proportion,” he continued. “This is neither good government nor good public policy.”

Touting his role in several audits of the Department of Education, Mr. Thompson counted down a laundry list of the department’s no-bid contracts, and announced that he would be reviving a failed state bill that would require all school contracts go through a competitive bidding process.

He also referred to his role as the principle investor of city pension funds, saying the school system would be considered a “risky investment” in the corporate world following the announcement of the third reorganization of the system in five years.

Following Mr. Thompson’s speech, Mr. Klein defended the department’s decision to reorganize the school system. He also dismissed the comptroller’s demands that the schools follow the same bidding procedures as other city agencies, noting that other school districts around the state are not required to do so.

“We’ve made changes, and we continue to look forward to working with him,” Mr. Klein said. “We got results. I think it’s time to build on it.”