Sources said an announcement of Kiffin's hiring is expected Monday, assuming there are no last-minute hangups, the sports network reported Friday.
The 33-year-old former Oakland Raiders coach will be taking his first job as head coach of a college team. He would succeed Phillip Fulmer, who coached his last game for the Vols Saturday after being fired early this month.
ESPN said Tennessee Athletics Director Mike Hamilton denied reports Wednesday that Kiffin had been offered a contract.
Kiffin was 5-15 in a season and a half with Oakland before being fired this season by team owner Al Davis.