Beauchamp made 428 appearances for Oxford across two spells and is widely regarded as one of their most talented ever players.
"The thoughts of everyone at Oxford United are with the family and friends of Joey Beauchamp who has passed away aged 50," read a statement from the League One club.
"The club will pay fitting tribute to one of its greatest ever players in due course but for now we ask fans to allow Joey's family some privacy at this incredibly sad time."
Beauchamp signed with his hometown club as a YTS trainee in 1987 and was sold to Premier League side West Ham in 1994 for a then club-record fee of Ł1.2m.
However, the winger did not play a competitive match for the Hammers, as he switched to Swindon just a couple of months later to be closer to his family.
Beauchamp returned to Oxford in November 1995, and ended up making 428 appearances and scoring 80 goals for United - leaving him 10th and fifth respectively in the club's all-time lists - before a toe injury ended his career in 2002.
Swindon and West Ham paid their respects to their former player.
Swindon said: "We are heartbroken to learn of the death of former Town player, Joey Beauchamp.
"Our thoughts and prayers go out to Joey's family and friends at this incredibly sad time."
Meanwhile, West Ham tweeted: "The thoughts of everyone at West Ham United are with the family and friends of Joey Beauchamp. Rest in peace, Joey."