WASHINGTON — The former acting F.B.I. director, Andrew G. McCabe, wrote a confidential memo last spring recounting a conversation that offered significant behind-the-scenes details on the firing of Mr. McCabe’s predecessor, James B. Comey, according to several people familiar with the discussion.

Mr. Comey’s firing is a central focus of the special counsel’s investigation into whether President Trump tried to obstruct the investigation into his campaign’s ties to Russia. Mr. McCabe has turned over his memo to the special counsel, Robert S. Mueller III.

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.