The striker has been on the books at Goodison Park since January 2018 but has struggled to get his Everton career going.
Tosun was sent out on a six-month loan spell to Crystal Palace in the 2019/20 season, while he returned to Besiktas on a temporary basis in the latter half of last term.
Even though Tosun only made four appearances across all competitions for the Turkish giants as his stint was cut short owing to a serious knee injury, it has been claimed they will look to sign him again in January.
However, according to Turkish daily Sabah, the Kara Kartallar have decided they will only swoop for Tosun next summer, when he will be available as a free agent.
The 30-year-old has stepped up his recovery but is expected to only be completely fit in February and Besiktas see signing him on a free transfer next summer rather than in January as the better course of action.
New Everton boss Rafael Benitez has roped in Salomon Rondon as a back-up for first choice Dominic Calvert-Lewin and also has Richarlison as another option in the centre-forward role, which could mean opportunities for Tosun will be limited even if he is fit and is available for selection after the turn of the year.
It remains to be seen if the Turk will opt to run down his contract at Goodison Park or try to push for a move away in January.