The 33-year old was trying to win a job as the Reds' fourth outfielder.
Bourgeois left the game against the Rockies on Saturday after a successful sacrifice bunt. He collapsed, holding his arm a third of the way down the first base line.
Manager Bryan Price said Bourgeois felt something in his shoulder in the Reds' first Cactus League game on March 2. He was held out a couple days and treated.
Brennan Boesch, who can play all three outfield positions, now has the inside track to win a spot on the Reds' bench.