Kiffin hasn't been officially offered the job but is acting as if he is the incoming coach of the Cougars, according to the newspaper.
Houston president Renu Khator is expected to sign off on the hiring in a meeting Thursday.
ESPN reports that no decision has been made on the hiring, but the school has narrowed in on three finalists -- Kiffin, Cougars offensive coordinator Major Applewhite and defensive coordinator Todd Orlando.
The 41-year-old Kiffin coached at Tennessee in 2009 before being hired to coach USC from 2010 until 2013. He is 35-21 as a collegiate head coach.
Kiffin also coached the Oakland Raiders to a 5-15 record from 2007 to 2008.
Former LSU coach Les Miles and Oklahoma offensive coordinator Lincoln Riley were also candidates for the job.
Herman left Houston after two seasons to become the coach at Texas on Nov. 27. The Cougars were 22-4 under Herman.