The Turkish international has struggled at Goodison Park and is not in the immediate plans of manager Carlo Ancelotti.
Tosun started a single game this season, the FA Cup tie against Rotherham.
Now it appears a return to Turkey and Besiktas could be on the cards.
A loan deal appears the most likely, with CSKA Moscow also interested.
"I would like to contribute to Besiktas until the end of the season.
"If Everton approves the transfer, it will not be a problem for me."