In the same statement, Mr. Lynch, for the first time, said the Police Department was reinstituting “quota policies and retaliation against police officers who fail to meet them,” an accusation that appeared designed to distract attention from the strife among his members. William J. Bratton, the police commissioner, said on Monday that the department had no productivity quotas.

Mr. Lynch, 51, is seeking his fifth four-year term as head of the union that represents 23,000 rank-and-file officers.

He faced a similar scene in 2011, when a delegate from a Bronx precinct stood up at a meeting and read a statement criticizing what the delegate characterized as Mr. Lynch’s lack of support for officers indicted after a three-year inquiry by Bronx prosecutors into ticket fixing. The delegate suggested Mr. Lynch consider stepping aside.

Mr. Lynch ran unopposed in that election. So far, no challenger has come forward this time, either.

But a union official, who requested anonymity in order to discuss the challengers’ election plans before they were publicly announced, said a slate of candidates would be announced in the coming days. “We’re the insurgents, I guess, until the time comes when we’re ready to announce,” the official said. “We’re running. It’s happening. Lynch knows it.”

The shouting on Tuesday came after a delegate with more than 20 years on the force, from the 75th Precinct in Brooklyn, challenged Mr. Lynch toward the end of the two-hour gathering, Michael Hernandez, a trustee, said.

“Why are we at war with the mayor?” the delegate asked, as Mr. Hernandez recalled. “Are we getting anything out of it?”

At that point, a shouting match ensued between dozens of supporters and opponents of Mr. Lynch.

Union officials said those who spoke out included Mr. Hernandez and another trustee from the Bronx, Joseph Anthony. Both men are under indictment in the ticket-fixing scandal. Brian Fusco and John Giangrasso, union trustees from Brooklyn, were also involved in the disagreement.