The 34-year-old Bourn, who was signed to a minor league deal Monday, was injured after Friday's workout. His status for Opening Day is unsure.
Bourn would earn $2 million should he make the Orioles' roster, a source told ESPN's Jim Bowden. The deal also included a March 25 opt-out clause if he's not on the roster, a source told ESPN's Jerry Crasnick.
Bourn, who first reached the major leagues in 2006, also played for Toronto, Philadelphia and Houston. His best season was 2011 when, while playing for the Braves and Astros, he set career-highs by batting .294 with 34 doubles and 61 stolen bases.
Last year, Bourn hit .264 with five homers, 38 RBIs and 15 stolen bases with the Orioles and Diamondbacks.