Tufan signed for Hull this summer from Besiktas and he has played 20 matches so far for the Tigers in the Championship.
The midfielder has scored four goals so far, with his equaliser against Sunderland after coming on as a substitute salvaging a point for Hull last weekend.
Besiktas are currently eighth in the Super Lig table and they are looking at ways to improve in the January window to be better in the second half of the season.
For that reason, Besiktas are looking at trying for Tufan on loan in January, according to Turkish daily Sabah.
However, Hull consider the Turkish star to be an important part of their squad and are thus not inclined to let him leave on loan.
In addition to being a former Besiktas star, Tufan has also played under their current coach Senol Gunes when they were both at Buraspor in the 2014/15 season.
However, with Hull not wanting to lose him in the January window, the prospect of him going to Besiktas in January is doubtful.
