The hitman parted ways with Besiktas in January 2018, joining Everton, but returned to the Turkish giants on a loan spell during the latter half of last season.
However, Tosun’s second spell at Vodafone Park did not go as planned as he could only feature in four games across all competitions, having suffered a serious knee injury.
The Kara Kartallar have identified their attacking department as one area that needs bolstering and want Tosun to return to the club for another spell as the January transfer window edges closer.
And according to Turkish outlet En Son Haber, Besiktas are keeping a close eye on Tosun, who has returned to the training pitch as he looks to regain his fitness.
Besiktas are monitoring Tosun’s progress as he makes his way back from injury with him expected to rejoin his team-mates in full training soon.
The Turk’s current contract at Everton expires next summer and Besiktas are also exploring the possibility of signing him, when he enters free agency.
Having added Salomon Rondon as a back-up for first choice Dominic Calvert-Lewin in the summer, and with Richarlison available as an alternative in the centre-forward position, it remains to be seen whether Tosun has any role to play under new Toffees boss Rafael Benitez if he stays at the club beyond January.