Royals manager Ned Yost told the Kansas City Star on Wednesday that a return for Hochevar by the end of May was the "best-case scenario."
The report said that Hochevar felt a twinge in his elbow during Monday's game against the Chicago White Sox.
The sprain was revealed in an MRI taken Tuesday. Hochevar will be shut down for at least two weeks.
Hochevar shifted to the bullpen last season, going 5-2 with a 1.92 earned run average and two saves in 58 relief appearances.