The reports goes onto state that van der Vaart has asked Dirk Kuyt and Wesley Sneijder about living in Istanbul.
The former Tottenham playmaker will be available on free transfer in June unless he renews his contract with HSV.
HSV are currently 16th in the German Bundesliga with 25 points after 26 games. Van der Vaart has scored five goals and provided two assists in all competitions for Hamburg this season.
Beşiktaş are currently second in the Süper Lig, just one point behind league leaders Galatasaray.
The Eagles will take on Başakşehir in the mini Istanbul derby after the international break