The Super Lig side are looking to bring in a right-back and are understood to have opened talks with the Gunners full-back's representatives.
Aksam reports that the Super Eagles are looking for cover for first-choice right-back Gokhan Gonul, who is now 34-years-old, and have made a sizeable offer to the 27-year-old.
Celtic have been linked with a number of defenders following the departures of Dedryck Boyata, Cristian Gamboa and Mikael Lustig, the end of Filip Benkovic's loan deal and the likely departures of Marvin Compper and Emilio Izaguirre, reducing the Hoops' first-team defensive options to Kristoffer Ajer, Jack Hendry, Jozo Simunovic, Kieran Tierney and Anthony Ralston.
It was reported on Friday morning that Arsenal, who are believed to be keen on Tierney, could attempt to use Jenkinson as a makeweight in a deal for the Scotland international.
According to Aksam, talks between the Turkish club and the player's agent are progressing well and the Besiktas are hopeful of striking a deal for the former England international.