Bostic replaces injured linebacker Reuben Foster, who suffered a season-ending torn ACL in his left knee Monday. The team placed him on injured reserve Wednesday.
Foster sustained the significant knee injury during the Redskins' first organized team activity session. Trainers tended to the linebacker on the field before putting his leg in an air cast.
Washington originally claimed Foster off waivers from the San Francisco 49ers in November 2018. The 49ers released him after a weekend arrest in Tampa on suspicion of misdemeanor domestic violence battery.
Florida prosecutors later dropped the charge. The NFL announced in April that Foster was fined two game checks, but wouldn't face suspension.
Bostic appeared in 16 regular-season games (14 starts) with the Steelers last season. He recorded 73 total tackles, 2.5 sacks and three passes defensed.
The Bears selected Bostic in the second round of the 2013 NFL Draft out of Florida. He has 313 combined tackles, 5.5 sacks, one interception and 10 passes defensed in his career, which included stints with the New England Patriots and Indianapolis Colts.