WASHINGTON — A former F.B.I. official at the center of the latest controversy over Hillary Clinton’s private emails acknowledged on Tuesday that an offer to swap favors with a State Department counterpart on an email classification issue had originated with him — until he realized the deal involved Mrs. Clinton and the 2012 attack in Benghazi, Libya.

“When I found that out, all bets were off; it wasn’t even negotiable,” the former F.B.I. official, Brian McCauley, said in a telephone interview.

Republicans have seized on the episode to accuse the State Department of trying to protect Mrs. Clinton, but Mr. McCauley’s account could undercut those attempts because he said he, not the State Department, had suggested the “quid pro quo.”

Mr. McCauley recounted in the interview that Patrick F. Kennedy, a senior State Department official, called him in spring 2015 looking for help in getting the F.B.I. to agree not to classify the disputed email. Mr. McCauley said he had agreed to try to help him if Mr. Kennedy would help him get the State Department to restore two spots that the F.B.I. had lost recently in the Baghdad embassy.