"It's got the potential of being a major injury," general manager Dave Dombrowski told Evan Drellich of NBC Sports Boston.
There is no timetable for Smith's return, and he needs a second opinion after undergoing an MRI.
The Red Sox announced Smith was placed on the 10-day disabled list with a right shoulder subluxation Tuesday, one day after allowing a home run to Khris Davis of the Oakland Athletics during a 6-5 loss.
In May 2016, the 28-year-old right-hander underwent Tommy John surgery. He's only pitched 23 2/3 innings since the Red Sox acquired him from the Seattle Mariners in December 2015.
Smith has been effective in limited action for the Red Sox, however, posting a 2.66 ERA and 1.35 WHIP in 29 appearances since 2016.