The deal opens playing top for one of the White Sox' top prospects, Jacob May.
May, 25, is batting .339 this spring and entered the 2017 Opening Day roster mix when Charlie Tilson went down with a stress reaction in his foot. Tilson isn't expected to play until June.
The Rays needed outfield help because Colby Rasmus is headed for the disabled list to start the season.
Bourjos, a Chicago native, signed as a free agent two weeks before spring training began and quickly entered a battle for the starting centerfield job. He hit .313 (15-48) with four doubles, three triples and three RBIs over 19 Cactus League games this spring.
Bourjos played the entire 2016 season with Philadelphia, hitting .251 (89-355) with 20 doubles, seven triples, five home runs, 23 RBI and 40 runs scored in 123 games.
The switch-hitting May was a third-round pick in the 2013 draft.