Flynn, who is the 18th-ranked prospect in the Royals' farm system, was fighting for the final job in the Kansas City bullpen with right-hander Ryan Madson, and the competition went down to the final weekend of Spring Training.
Royals officials do not believe Flynn's injury will be season-ending, and they estimate he could be back in August. The injury does not require surgery.
Flynn, who will turn 25 on Sunday, was acquired from the Marlins in the Aaron Crow trade. Crow recently was diagnosed with a partially torn ulnar collateral ligament in his elbow.
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.