According to the report, the team holds an option for a fifth year. Terms of the contract were not available, but Smith is expected to make more than$5 million a year.
Smith replaces Greg Schiano, who was fired on Monday after two years on the job and three years left on his contract.
Smith, 55, had been out of coaching since being fired after the Chicago Bears finished 10-6 and missed the playoffs in 2012. He had a 81-63 record in nine seasons with the Bears and led the team to three division titles and one Super Bowl appearance.
Smith was also the Buccaneers linebackers coach from 1996-2000.
He reportedly also interviewed with Houston Texans for their head coaching job. The Texans ended up hiring Penn State head coach Bill O'Brien. BURKE: Top-10 Mock Draft: Projections for 10 worst teams after Week 17