CNN Turkey are reporting that Besiktas are leading the race to sign the 22-year-old, but Celtic, Nice and Feyenoord are also interested.
Bosnian-born Babic has represented Serbia at youth international level from under-17s to under-21s.
Celtic are likely to see both Dedryck Boyata and Filip Benkovic leave the club this summer. Boyata is in the final year of his contract while Benkovic will return to parent club Leicester City at the conclusion of his loan deal.
The Scottish champions are also said to be pursuing Chelsea defender Tomas Kalas, but will face a fight with Bristol City desperate to hold on to their loan player.