The 32-year-old is widely expected to leave the Miners in the winter transfer window having been told he will not be playing for Schalke again.
And with his current deal due to expire at the end of the season, chances are Jones will be available on a cut-price deal.
According to Bild, he is currently undergoing a medical at Besiktas and is expected to join the Black Eagles in the coming days.
While the winter transfer window closes tomorrow in most European countries, Turkish clubs have until 3rd February to sign new players, meaning there is still plenty of time for the deal to go ahead.
Jermaine Jones has spent last six and a half years in Gelsenkirchen, making a total of 129 Bundesliga appearances and scoring 7 goals. The USA international also spent six months on loan at Blackburn Rovers.