In an interview on Thursday, Mr. Peters disputed City Hall’s interpretation of the letter.

“It certainly wasn’t a threat,” Mr. Peters said, “both because the fact of the investigation going on was already well known to City Hall, and I don’t believe that anyone would seriously think that what we would do in our investigation would in any way be influenced by what City Hall does.

“We don’t permit noninvestigatory concerns to influence it.”

Mr. Peters said that he could not confirm a continuing investigation, including whether or not his office was investigating the yeshiva review, and suggested that more than one inquiry may be underway.

“If they believed this was solely a reference to any investigation involving yeshivas, that belief would be wrong,” he said.

There is no indication that Mr. Peters’s investigators have found evidence that City Hall interfered with the yeshiva review, but the mere existence of an inquiry could complicate matters as Mr. de Blasio considers whether to fire Mr. Peters over the whistle-blower case.

No mayor has ever fired an investigations commissioner, and doing so could open Mr. de Blasio to the charge that he was seeking to quash a politically sensitive investigation that has put his inner circle under scrutiny.

The Department of Education began a review of the yeshivas in 2015, after complaints that the schools were not providing adequate education in nonreligious subjects, including math, science and English. The review, completed in August, concluded that 15 schools were either meeting state educational requirements or instituting improvements. Another 15 schools refused to cooperate, according to the Education Department.

The review was criticized for its slow pace, its conclusions and the city’s inability to obtain the cooperation of many of the schools.