The Black Eagels held talks with the 30-year-old earlier this week according to HaberTürk. However, Vlaar is keen to stay in the Premier League and has ended talks with Besiktas.
The Netherlands international has also been closely linked with Southampton, Napoli and Galatasaray.
Vlaar who joined Villa for €5 million from Feyenoord, is free to talk directly to other clubs with his contract set to run out at the end of the month.
The 30-year-old provided two assists in 23 appearances for Aston Villa in all competitions this season.
Beşiktaş finished the Turkish Super Lig season in third place and will play in the Europa League next season.