The report states that Galatasaray have offered a lucrative €3 million deal for the 31-year-old whose contract set to expire on 30th June 2015.
Reyes came close to a move to Galatasaray back in 2011 before joining his favourite club Sevilla.
The 31-year-old has seven assists and four goals in 35 appearances for Sevilla in all competitions this season. The Andaluz club sit in 5th place with 73 points in 37 games, one point behind 4th placed Valencia.
Sevilla know a victory this weekend could see them qualify to the UEFA Champions League if Valencia drop points against an already relegated Almeria.