The Turkey international spent the latter half of last season at Besiktas, but had to cut his loan stint short owing to a serious knee injury.
Tosun arrived at Goodison Park in January 2018 from Besiktas, however he never managed to find his rhythm in Everton colours and his deal is set to expire at the end of this season.
The Turkish giants want the striker to join their ranks for a third stint, and are keen to bring in him in January as a backup for Michy Batshuayi.
However, according to Turkish daily Takvim, the Besiktas hierarchy are panicking at the moment with Everton boss Rafael Benitez including Tosun in his last two Premier League matchday squads.
Having returned from injury, Benitez has had Tosun at his disposal in recent outings and he could be handed some game time in the upcoming games.
Besiktas are hoping to complete a swoop for the striker in January and are now desperate to sit at the negotiating table with Everton as soon as possible as there is a possibility that the Merseyside giants could opt to keep him for the remainder of the season.
With the addition of Salomon Rondon in the summer, Benitez now has the Venezuelan along with Richarlison to step up in the absence of first choice Dominic Calvert-Lewin, and it remains to be seen whether Tosun has any part to play in his plans.