The Ivory Coast international made the switch to the Emriates Stadium in a deal that was allegedly worth in the region of £72m.
Pepe has since contributed 27 goals and 21 assists from his 112 appearances in all competitions, yet it has been perceived that he has never lived up to his price-tag.
As a result of falling down the pecking order under Mikel Arteta, Pepe spent last campaign on loan at Nice, where he chipped in with eight goals and one assist from his 28 outings.
Nevertheless, with just a year remaining on his contract, there was never any prospect of the 28-year-old being reintegrated into Arsenal's first-team squad.
The Gunners have struggled to attract suitable offers, though, and the decision has seemingly been made to accept any reasonable offer for Pepe in an effort to reduce their wage bill.
According to L'Equipe, Besiktas have formally approached Arsenal over securing a deal for Pepe in the coming days.
With terms having allegedly been agreed between the two teams, Pepe is now in line to undergo a medical in Paris.
Should a deal go through, Arsenal would be boosting their position with FFP having shelled out over £200m on three players before the beginning of the campaign.
In return, Arsenal have generated in the region of £80m through player sales while allowing a number of other players to go out on loan.
Besiktas have already made a number of eye-catching signings this summer, including former Liverpool midfielder Alex Oxlade-Chamberlain and AC Milan winger Ante Rebic.
A total of seven points have been accumulated from three matches in the Super Lig, while Besiktas have secured a place in the Europa Conference League group stages through overcoming Dynamo Kiev in the playoff round.