The climax came on Thursday afternoon at a joint meeting of the board’s executive and audit committees, when Mr. Perelman told his fellow trustees that he believed some of the laws governing nonprofits were not being followed at Carnegie and expressed frustration that no investigation into his concerns had been initiated, according to someone familiar with the proceedings, who was granted anonymity to describe what happened at a confidential meeting.

Mr. Perelman criticized board members for placing “a premium on avoiding tension and disagreement,” the person said, and told them that while he would serve out his term, he would not run for re-election as chairman at the board meeting next month. The board then agreed to hire a lawyer to examine his concerns, which involved how Carnegie vetted what Mr. Perelman called “related-party transactions” in looking at deals that posed potential conflicts of interest.

Carnegie Hall officials did not return calls seeking comment on Thursday.

One member of Carnegie’s board said Mr. Perelman might have decided to step aside next month because he was unlikely to be re-elected. The board member, who spoke on the condition of anonymity, said the trustees had felt blindsided this summer when Mr. Perelman briefly suspended Mr. Gillinson. The board member added that while they would investigate the concerns he raised, they did not believe that Carnegie had governance or transparency problems, and denied that the board had been dragging its feet on hiring a lawyer to investigate.

“The message from the Carnegie side is Clive Gillinson was an exemplary executive director for 10 years, we believe people are innocent until proven guilty, and we are quite confident that our governance is subject to perfect scrutiny,” he said.

Mr. Perelman’s reputation as a sharp-elbowed businessman with a penchant for taking on battles — he has sued the investment bank Morgan Stanley and the celebrity art dealer Larry Gagosian, among others — preceded his rise to chairman of Carnegie’s board. But he also has a long history of philanthropy, has been a trustee at Carnegie since 1988 and has donated roughly $30 million over the years to the hall. Its main stage is named for him.