Goodwin became a free agent Friday after declining an outright assignment to the minors.
Goodwin (back) isn't included on the White Sox's ALDS roster, Bruce Levine of 670TheScore.com reports.
The White Sox placed Goodwin on the 10-day injured list with lower back spasms Sunday.
Goodwin may not be available off the bench Saturday at Cleveland due to back tightness, Daryl Van Schouwen of the Chicago Sun-Times reports.
Goodwin will serve as the White Sox's designated hitter and will bat ninth Monday against the Tigers.
Goodwin is not starting Tuesday against the Angels, Vinnie Duber of NBC Sports Chicago reports .
Goodwin is on the bench for Sunday's game against the Red Sox.
Goodwin will sit Thursday against the Athletics, Vinnie Duber of NBC Sports Chicago reports.
Goodwin replaced an injured Eloy Jimenez (knee) and went 1-for-2 with two RBI in Tuesday's 6-3 win over Oakland.
Goodwin isn't starting Tuesday's game against the Athletics.