He replaces Dick Pole, who was fired on Oct. 2 as the only change on manager Dusty Baker's staff.
Price has spent the last 10 years as a pitching coach in the majors. He was with Arizona from 2006 until May 8, when he resigned after manager Bob Melvin was fired. Price also spent six seasons with the Seattle Mariners.
Price was a pitcher in the Angels' and Mariners' minor league systems before he started coaching with Seattle.
The Reds are coming off their ninth straight losing season, their worst stretch in a half-century.