The Northern Ireland international striker, a key figure in the club's promotion push was offered a one-year deal by Paul Sturrock at the end of the season.
But despite knowing an offer was on the table, Quinn hesitated after nothing was sorted out and reluctantly decided to look elsewhere.
Cheltenham is only a 45-minute drive from his home in the midlands and it is believed that that Town will offer the former Willem II star a two-year deal.
Quinn, who has been training with Birmingham City, has been offered deals by other clubs and considerably less by Town but Cheltenham boss John Ward is hoping he can sign the experienced forward before the start of the season.