The first year of the deal would be worth $27 million guaranteed, roughly the same amount Winston would receive on a franchise tag, and the second year would be a team option, according to Laine.
Winston is scheduled to become an unrestricted free agent in March after five turbulent seasons as the Buccaneers' franchise quarterback.
ESPN's Jeremy Fowler also reports people around the league expect the Bucs to use their franchise or transition tag on Winston or sign him to a modest, short-term deal.
Buccaneers head coach Bruce Arians said in mid-February the team will not make a determination on what it will do at quarterback without seeing how the other free-agent dominoes fall.
"You're just sitting there waiting to see is there someone available? And is he a better option," Arians said. "That's the problem. You get about three days to decide - the legal tampering (period). Is somebody else gonna get let go? What's the trade values? It's really monotonous right now because you don't really have any answers."
Winston has yet to lead Tampa Bay to a playoff berth since being selected first overall in the 2015 draft.
The Buccaneers had the No. 1 aerial attack in 2019 and Winston led the league in passing with 5,109 yards. He finished second in passing touchdowns with 33, but also led the league in interceptions by a substantial margin with 30, recording the league's first 30-30 season by a quarterback.