Sampson made his major league debut last season before surgery to repair of flexor tendon in his right elbow. He is expected to be in an injury rehabilitation program for the start of the 2017 season.
Sampson was on Seattle's 60-day disabled list, and has been transferred to the Rangers' 60-day DL.
The 25-year-old Sampson started June 18 for the Mariners at Boston. He didn't make his next scheduled start five days later because of the injury and then had surgery June 30.
Sampson was a fifth-round pick by Pittsburgh in the June 2012 draft. The Pirates traded him to Seattle in July 2015 in a deal involving J.A. Happ.