Blackburn (elbow) signed a minor-league contract with Cincinnati on Friday, Brandon Warne of ZoneCoverage.com reports.
Blackburn (elbow) was outrighted from the 40-man roster Friday and is expected to elect free agency.
Blackburn (elbow) was moved to the 60-day-disabled list Monday.
Blackburn was diagnosed with a strained UCL on Thursday, Evan Grant of The Dallas Morning News reports.
Blackburn (elbow) is not expected to return until after the All-Star break, Evan Grant of The Dallas Morning News reports.
Blackburn exited Monday's spring game early due to right elbow tightness.
Blackburn was optioned back to Triple-A Round Rock following Wednesday's game against the Mariners, Stefan Stevenson of the Fort Worth Star-Telegram reports.
Blackburn was recalled from Triple-A Round Rock on Monday, Jeff Wilson of the Fort Worth Star-Telegram reports.
Blackburn was traded to the Rangers on Sunday.
Blackburn was designated for assignment by the Giants on Tuesday.