Reeves started his career as an apprentice with A.F.C. Bournemouth. He made his League debut in the 1974–75 season, and played 63 games before John Bond signed him for Norwich City in January 1977 for a fee of £50,000. After 133 games and 42 goals in all competitions, he earned Norwich their first £1 million fee when he was transferred to Manchester City in March 1980 for £1.25m. Reeves scored 39 goals from 158 appearances in all competitions, including a penalty in City's defeat at the hands of Tottenham Hotspur in the 1981 FA Cup Final replay. He subsequently played for Burnley, continuing his trend of playing under John Bond, but was forced to retire from the professional game at the age of 26 because of a hip injury.