Griese opened last season 5-1 as the Buccaneers starter, but suffered a torn ACL and MCL on Oct. 16 that landed him on injured reserve. The six-foot-3, 214-pounder is expected to be ready by training camp.
Griese has started 72 games over the course of his eight-year career and is clearly the most accomplished quarterback on the roster. He's completed 63.1 percent of his passes for 16,344 yards with 103 touchdowns and 78 interceptions and an 84.8 passer rating.
Griese, who turned 31 on Saturday, will push Kyle Orton to third on the depth chart. Before making his decision, Griese met with the Bengals but felt comfortable enough to sign with the Bears after visiting Halas hall on Sunday and then having a face-to face conversation with offensive coordinator Ron Turner on Monday.