Hochevar, the first pick in the 2006 amateur draft, got hurt Monday while pitching to Matt Davidson, his final batter during a two-inning outing against the Chicago White Sox. The sprain was revealed in an MRI taken Tuesday.
Manager Ned Yost said Wednesday that Hochevar won't throw for two-to-three weeks and "probably the best-case scenario" would be a return at the end of May or early June.
"His second-to-the-last pitch against the White Sox he felt a twinge in his elbow," Yost said. "He ended up throwing a cutter to strike the guy out. He felt OK coming in, but the next day it was real stiff and sore."
Hochevar was 5-2 with a 1.92 ERA and two saves in 58 relief appearances last year. He is earning $5.21 million and is eligible for free agency after the season.
He was sidelined from mid-June until mid-September in 2010 because of a sprained elbow and made only 17 starts and one relief appearance that year.
"We were a little nervous that it could have been worse," Yost said. "We think we got as good a report as we'd hoped for coming back. But, again, you really won't know until three weeks to see exactly where we're at."
Hochevar was a candidate for the final rotation slot this season.