The 2018 second-round pick had a season-high 52 receiving yards on three catches during Sunday's overtime loss to the Baltimore Ravens. He has nine catches for 140 yards on the season.
NFL Network first reported that Washington is expected to miss multiple weeks.
Beyond the loss of quarterback Ben Roethlisberger for the season, the Steelers are also without running back Jaylen Samuels and have uncertainty at quarterback with Mason Rudolph in the concussion protocol.
Samuels will miss "roughly a month" after undergoing arthroscopic knee surgery Monday, coach Mike Tomlin said Tuesday. Rudolph entered the concussion protocol Sunday after a hit by Ravens safety Earl Thomas. He was at the Steelers' facility Monday and talked with Tomlin.
Washington injured his shoulder while catching a 26-yard pass on the play when Rudolph was drilled by Thomas.
ESPN's Brooke Pryor and The Associated Press contributed to this report.