Logan has decided not to have surgery after consulting with doctors. He was injured throwing a pitch against San Francisco on July 19 and diagnosed with a high grade latissimus strain.
Logan, who is on the 60-day disabled list, is 1-0 with a 4.71 ERA in 38 appearances. He signed a one-year contract with a club option for 2018 in February.
Cleveland's bullpen has been hit hard by injuries. Left-hander Andrew Miller was placed on the 10-day disabled list Wednesday with right knee tendinitis, leaving manager Terry Francona without the dominant All-Star.
The Indians signed veteran left-hander Craig Breslow to a minor league contract and assigned him to Triple-A Columbus. He appeared in 30 games for Minnesota and was released last month.