Cobb passed his physical with the Tigers on Tuesday, Jon Heyman of the New York Post reports.
The Tigers have agreed to a one-year contract with Cobb, Ken Rosenthal of The Athletic reports.
The Guardians removed Cobb from their ALCS roster due to an acute left low back strain.
Cobb has been named the starter for Game 1 of the American League Championship Series versus the Yankees, Mandy Bell of MLB.com reports.
Guardians manager Stephen Vogt said Monday that Cobb will start Wednesday in Game 3 of the ALDS in Detroit, Jesse Rogers of ESPN.com reports.
Cobb (finger) is included on the Guardians' ALDS roster.
The Guardians placed Cobb on the 15-day injured list Thursday with a right hand blister, Zack Meisel of The Athletic reports.
Cobb will not make his scheduled start Saturday due to a blister, Joe Noga of The Cleveland Plain Dealer reports.
Cobb (2-1) earned the win Sunday, allowing one run (zero earned) on two hits over six innings against the Pirates. He struck out six.
The Guardians reinstated Cobb (finger) from the 15-day injured list to start Sunday against the Pirates, Zack Meisel of The Athletic reports.