The Reds will receive infielders Jose Peraza and Brandon Dixon and outfielder Scott Schebler from the Dodgers. The White Sox send pitcher Frankie Montas, infielder Micah Johnson and outfielder Trayce Thompson to the Dodgers.
The 29-year-old Frazier, who likely pushes new White Sox infielder Brett Lawrie from third base to second, batted .255 with 35 homers and 89 RBIs. He won last summer's All-Star Home Run Derby in Cincinnati and was an NL All-Star in each of the last two seasons.
"Todd is one of the finer all around third basemen in the league. He is a high-character, smart baseball player who will provide an impact bat in the middle of our order," White Sox general manager Rick Hahn said in a statement. "He solidifies us defensively at third base and provides a great presence in our lineup and in the clubhouse."
Frazier has a $7.5 million salary next year and can become a free agent after the 2017 season.
According to MLB.com, Peraza was the Dodgers' fourth-ranked prospect last season, while Schebler was 13th. Montas was the White Sox's third-ranked prospect, while Johnson was fifth and Thompson 14th.
Montas, a 6-foot-2 right-hander, struck out 20 in 15 major league innings last season. A hard thrower, he likely will start the season in the rotation at Triple-A Oklahoma City, but the Dodgers think he could be an impact starter in the majors as soon as 2016.
ESPN Staff Writer Mark Saxon and The Associated Press contributed to this report.