Puk was shut down after experiencing biceps soreness that eventually went to his forearm. Dr. James Andrews recommended the elbow surgery.
Puk, 22, was the sixth overall pick in the 2016 draft out of the University of Florida. He struck out 184 batters in 125 innings split between Class A and Double-A last season, going 6-10 with a 4.03 ERA.
He pitched well this spring, holding opposing hitters to a .184 average in 10 innings, and he was thought to have a chance to make the club.
But he did not pitch again after giving up four runs in his last outing on March 15 and feeling tenderness in his left arm.
Puk was ranked 13th in ESPN's Keith Law's list of top 100 prospects for 2018.