In 75 games this season, the 25-year-old Jimenez batted .255 with seven home runs, 26 RBIs, 35 runs and five triples.
Jimenez was the starting second baseman for the Sox since Ray Durham was traded in July of last season, but his base-running lapses and key defensive miscues landed him on the bench and put the White Sox in the market for Alomar.
Rios, 31, was signed on as a minor league invite in the off-season, making the team as the backup outfielder. With the trade for Everett and recent play of both Aaron Rowand and Willie Harris, Rios was the odd man out.
Rios batted .222 with two homers and 11 RBIs in 42 games and 99 at bats for the White Sox. Rios has a .271 average for his career, which has included stops in San Francisco and Pittsburgh.