Ian Rapoport of the NFL Network reported that the deal is worth $25.35 million over four years and includes a $16.7 million signing bonus.
Winston also will receive roster bonuses on the fifth day of training camp in 2016, '17 and '18, Rapoport reported.
Over two seasons as the Florida State's starter, Winston completed 66 percent of his passes for 7,964 yards with 65 touchdowns and 28 interceptions. He won the Heisman Trophy in 2013.
Though Winston dealt with a number of off-field issues while at Florida State, Bucs coach Lovie Smith said before the draft that the team was comfortable with Winston as the face of the franchise. Winston was accused of sexually assaulting a former Florida State female student in December 2012. Authorities did not file charges, and Winston was also cleared in a university Code of Conduct hearing.
In addition, Winston was cited for stealing crab legs from a grocery store in Tallahassee and was given a one-game suspension after he was witnessed shouting an obscene phrase in a Florida State student union.
SI.com's Doug Farrar assigned the Bucs a B for picking Winston.