Rainey tossed two scoreless innings of relief in Monday's 4-1 loss to the Rays. He walked two and struck out two.
Valdez (8-9) took the loss against the Rays on Monday, allowing four runs (three earned) on eight hits and two walks in six innings. He failed to record a strikeout.
Lee will start at second base and bat second in Monday's game against the Rays.
The Tigers optioned Madden to Triple-A Toledo on Monday, Cody Stavenhagen of The Athletic reports.
The Tigers designated Vanasco for assignment Monday, Cody Stavenhagen of The Athletic reports.
The Tigers signed Kinley to a one-year contract, Cody Stavenhagen of The Athletic reports.
Torres is out of the lineup for Monday's game against the Rays.
Greene (hamstring) returned to Detroit on Monday and will take part in conditioning work at Comerica Park prior to the Tigers' series opener versus the Rays, Evan Woodbery of MLive.com reports.
Estevez (shoulder) threw in a game at the Royals' Arizona complex on Saturday, Joel Penfield of KC Sports Network reports.
Gonsolin (elbow) is on track to start a rehab assignment with Triple-A Omaha this week, per MLB.com.
