The team announced that Price, 33, was treated with a shot of cortisone in the wrist Wednesday.
"The hope is for him to come [back] sooner rather than later, but we don't have a timetable," manager Alex Cora said about Price, who was not made available to address his injury. "We'll make sure he's OK, and we'll go from there."
Cora noted that the ailment, classified as a triangular fibrocartilage complex (TFCC) cyst, was not cancerous or related to any other serious illness.
Price most recently pitched Sunday against the New York Yankees after being reinstated from paternity leave. He allowed seven runs on nine hits in 2⅔ innings in New York, falling to 0-3 with a 10.59 ERA in his past four starts.
Cora said Price told him after the game that he "didn't feel 100%" and "felt tight."
"His last three or four [starts], as far as the location and the action of the pitches, it wasn't there," Cora said. "Regardless if he's throwing 87 or 95 [mph], precision and strikes is his game."
Price missed 14 games earlier this season with left-elbow tendinitis and was plagued by elbow inflammation during the 2017 season. He is 7-5 with a 4.36 ERA over 21 starts this season.
"Hey, at least we found out," Cora said. "Whatever people were thinking about if it's the elbow or something else, at least we know [it's not]. We'll shut him down for a little bit, and then he'll be back."