Baseball's sixth-leading career home run hitter with 608 was the American League's 1997 MVP while playing for the Seattle Mariners and is a 13-time All-Star looking to play in a World Series for the first time in his 20 major league seasons.
With the White Sox, he joins a team with a 1 1/2-game lead over Minnesota Twins for first place in the AL Central.
He hit his 600th home run earlier this season.
A source told MLB.com that the Reds received right-handed pitcher Nick Masset and second baseman Danny Richar.
Masset, 26, has appeared in 32 games this season, compiling a 1-0 record with a 4.63 ERA.
Richar, 25, has batted .230 with six home runs and 15 RBI in 2007, his rookie season. He has spent the better part of six seasons in the minors, most recently with Triple-A Charlotte.