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. Yost said Wade Davis, Danny Duffy and rookie Yordano Ventura are "realistically fighting for that spot" and Hochevar would pitch from the bullpen when he returns.