Mrs. Gresser has lately sought to mend fences with the Chancellor, who has emerged as a potent political player and valuable mayoral ally.

But the law limits him to two consecutive terms, a complication that has sent mayoral allies in search of legal loopholes and nuance, which they have found.

"I think Mike has done a pretty good job," said one highly regarded chief executive and mayoral ally, who spoke on the condition of anonymity so as not to alienate City Hall.

"When it comes to the mayor you cannot function with blinders on," said Jerry Cammarata, the board member from Staten Island and a mayoral ally.

And there is the Yonkers City Council, where a veto-proof majority of mayoral allies has long stood in the way of more equitable spending.

You cannot afford to hire a lobbying firm of a mayoral ally like Raymond B. Harding, so what is a community to do?

When Mr. Kiley emerged as a candidate on Friday, mayoral allies began pressuring Ms. Thomson to back him, a board official said.

They are already deeply familiar with the record of Gifford Miller, who has alternately been a mayoral ally or foe as speaker of the City Council.

There were investigations into cronyism and kickbacks, and two convictions of mayoral allies for fraud.

Yet he acknowledged that one of the reasons he gave then - the possibility that a mayoral ally would secure more city money for public schools - had not materialized.