Wells hasn't pitched in the major leagues since 2009 for Cincinnati, his eighth team over 11 seasons in a career that started with the White Sox from 1999-2001.
The deal was announced Saturday, two weeks before Wells' 35th birthday.
Chicago picked Wells 16th overall in 1998 out of Baylor. He made his major league debut the following year going 4-1 with a 4.04 ERA in seven starts.
His career record is 67-99 with a 4.71 ERA in 289 games for the White Sox, Pittsburgh, Texas, St. Louis, Colorado, Kansas City, Washington and Cincinnati.