The Black Eagles are reported to have made an enquiry for the out-of-favour forward and are considering the Italian international as an option should Mario Gomez decide to leave over the summer.
Gomez scored 28 goals last season – the highest ever for a foreign players at Besiktas – in the Istanbul giants title run.
The Germany international is however, on loan and his impressive form last season has attracted a lot of interest from the Premier League and Bundesliga.
Liverpool have transfer listed Balotelli and want to offload him to free up the £80,000-a-week wages that he is currently on but Galatasaray and Besiktas would struggle to cover his current wages.
Balotelli has just returned from a disappointing season long loan at AC Milan scoring just one league goal in 23 games.