Tosun arrived at Goodison Park in January 2018 in a big-money move, but has struggled to live up to his billing in England.
The striker’s current deal at Everton is set to expire in the summer, and he has largely been on the fringes of the team this season, with niggling injuries also forcing him on to the sidelines at times.
Besiktas were keen on bringing Tosun back to Turkey in the forthcoming window and the player is also claimed to be open to the move.
However, according to Turkish daily Sabah, the Kara Kartallar are now having second thoughts about snapping up Tosun, given his injury issues.
The Besiktas hierarchy are undecided over a summer swoop for Tosun, who is currently working his way back from a quadriceps injury and is expected to be fully fit only in July.
Tosun still shares a good relationship with Besiktas president Ahmet Nur Cebi, and a move back to the Super Lig side is not ruled out, but they are yet to decide whether to take a risk and sign him.
The Turk is unlikely to have a future at Everton beyond this season and it remains to be seen where he ends up playing next term.
