The report states that the Black Eagles are preparing a €2 million per season deal for the experienced Spanish international whose contract set to expire on 30th June 2015.
The Turkish newspaper goes onto state that Reyes is looking for a deal in the region of €3 million per season.
Besiktas are looking to add experience to a young squad ahead of the 2015-16 season.
Reyes 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.