“While the mayor respects Stan’s opinion and thanks him for his unrivaled legacy of public service, you don’t gamble or play a game of chicken when it’s your job to prevent Donald Trump from becoming the landlord of 400,000 public housing residents,” Mr. Phillips said.

A HUD spokesman did not respond to a message seeking comment.

The agreement stops short of a full federal takeover of the city’s housing authority, something Mr. de Blasio opposed. In the announcement with Mr. Carson, Mr. de Blasio called it “a strong path forward and a very tangible path forward.”

It includes the appointment of a powerful federal monitor — paid for by New York City — and a commitment from the city to spend an additional $2.2 billion over 10 years on repairs for its crumbling system of 175,000 public housing apartments. It also gives some authority to the federal housing secretary that is similar to a federal receiver, such as the ability to abrogate city contracts and go around Civil Service requirements.

“This is a receiver in everything but name,” Mr. Brezenoff said.

Because he refused to sign the deal, the general manager of Nycha, Vito Mustaciuolo, did so in his place. On Tuesday, City Hall announced that Mr. Brezenoff would be replaced as interim chairman of Nycha by the middle of the month. The city’s sanitation commissioner, Kathryn Garcia, was appointed to take over temporarily as officials hunt for a permanent leader.

When Mr. de Blasio named Mr. Brezenoff to lead the troubled public housing authority in April, he did so amid scandal over its failure to properly inspect for lead, and in the middle of a sprawling federal investigation.

Mr. Brezenoff, who by that point had already become the go-to Mr. Fix-It for Mr. de Blasio, took over weeks before the city and the authority moved to settle a case brought by the United States attorney, Geoffrey S. Berman, over dangerous conditions and endemic mismanagement at the authority.

Mr. Brezenoff, 81, said that he had felt some hope late last year after a federal judge rejected the terms of the initial deal between Mr. de Blasio and the United States attorney’s office. The judge, William H. Pauley III, strongly suggested that the federal government should take over New York City public housing, even as he chastised federal officials for abdicating their legal responsibility to the residents of the nation’s largest public housing system.