In the document, whose contents have not been previously reported, Mr. McCabe described a conversation at the Justice Department with the deputy attorney general, Rod J. Rosenstein, in the chaotic days last May after Mr. Comey’s abrupt firing. Mr. Rosenstein played a key role in the dismissal, writing a memo that rebuked Mr. Comey over his handling of an investigation into Hillary Clinton.

But in the meeting at the Justice Department, Mr. Rosenstein added a new detail: He said the president had originally asked him to reference Russia in his memo, the people familiar with the conversation said. Mr. Rosenstein did not elaborate on what Mr. Trump had wanted him to say.

To Mr. McCabe, that seemed like possible evidence that Mr. Comey’s firing was actually related to the F.B.I.’s investigation into the Trump campaign’s ties to Russia, and that Mr. Rosenstein helped provide a cover story by writing about the Clinton investigation.