Gonzalez was optioned to Triple-A Charlotte on Monday.
Gonzalez went 0-for-4 with three strikeouts in Saturday's 8-4 win over the Tigers.
Gonzalez was outrighted to Double-A Birmingham on Sunday, Colleen Kane of the Chicago Tribune reports.
Gonzalez was acquired by the White Sox on Saturday.
Gonzalez was designated for assignment by the Astros on Saturday, Jake Kaplan of the Houston Chronicle reports.
Gonzalez was optioned to minor league camp Thursday.
Besides starter Jason Castro, Gonzalez is the only other healthy catcher on Houston's 40-man roster.