The statement by Mr. Carr, the special counsel’s spokesman, was unusual because it appeared to be the first time he had publicly challenged the facts of a news media account that had generated significant attention for its revelations about the president.

The New York Times has not independently confirmed the BuzzFeed report. One person familiar with Mr. Cohen’s testimony to the special counsel’s prosecutors said that Mr. Cohen did not state that the president had pressured him to lie to Congress.