"It's just unfortunate," Nagy said. "Bryce has shown me a lot. He's very valuable. That's a big loss for us."
Callahan, 27, emerged over the past three years as a valuable member of the Bears' secondary. He started 10 games this season, recording 45 tackles, two sacks, two interceptions and six pass breakups.
An undrafted free agent out of Rice in 2015, Callahan also picked off two passes last season.
Veteran Sherrick McManis is the leading candidate to fill in for Callahan in Chicago's defensive backfield.
McManis, annually one of the Bears' best performers on special teams, has appeared in 96 games since joining the team in 2013, but has only five career starts on defense.
The Bears have a chance to clinch the NFC North with a win over the Green Bay Packers on Sunday at Soldier Field. However, to punch their ticket to the playoffs for the first time since 2010, the Bears have to slow down Packers quarterback Aaron Rodgers, who passed for 286 yards and three touchdowns in Green Bay's comeback victory over Chicago in Week 1.
"Bryce, man, I wish he didn't get hurt, but that's the game sometimes," McManis said Wednesday. "So I need to do my best that I can and go out there and have some fun.
"I've got an opportunity I want to take advantage of -- to go out there and help the team get a big win because it's the next game."
The Bears also signed defensive back Marcus Williams to provide additional depth in the secondary. Williams started 14 games for the New York Jets from 2014 to '16 and appeared in two games for the Bucs this season.