The journeyman quarterback will make $4.75 million in 2014-15 and $5.25 million in 2015-16, the sources told Schefter. The deal also guarantees $4.75 million, a source told ESPN.com's Michael C. Wright. McCown's agent, Michael McCartney, confirmed the two-year deal.
He can earn $5 million more in incentives -- $1 million each year for playing time and another $3 million for reaching the playoffs, sources told Schefter.
Bucs coach Lovie Smith said Wednesday that McCown will compete for the starting job.
"Every position there's competition. We have starting rotations," Smith said. "Gerald McCoy is at the top of the depth chart with our starting rotation and then you have to have a certain level of play for guys to stay that way. That's the way it's always been with me. There has to be a starting spot, a starting point, a starting person to go out there first and that would be Josh."
McCown, 34, played a significant role for the Chicago Bears as they implementing a new offense under coach Marc Trestman and offensive coordinator Aaron Kromer, and he has been described by Bears general manager Phil Emery as a "glue guy" in the locker room.
He played in a total of eight games, with three wins in five starts filling in for injured starter Jay Cutler, with his strong performances stirring debate about whether he should become the starter on a full-time basis.
McCown set a team record with 36 completions in a Nov. 24 loss to the St. Louis Rams, was named NFC Offensive Player of the Week on Dec. 15, and ranked third in the league with a passer rating of 109.0.
ESPN.com Bears reporter Michael C. Wright contributed to this report.