The Ivory Coast international has supposedly been told by Mikel Arteta that he will not play for the club again, four years on from his £72m arrival from Lille.
While clubs in England can no longer sign players, the Turkish transfer window is open until September 15, while the Saudi Arabia market does not close until later on Thursday.
A switch to the Middle East has unsurprisingly been mooted, but Besiktas have apparently emerged as the frontrunners to sign the exiled 28-year-old, who is already said to have undergone a medical.
However, a fresh report from L'Equipe states that Pepe's switch to the Super Lig is now on hold, as Arsenal and Besiktas have restarted negotiations following a prior disagreement.
The French outlet states that Pepe passed his medical on Monday without any problems, but his transfer to Besiktas will be blocked until the two clubs can come to a full agreement.
With the Turkish transfer window not closing for another eight days, there is still optimism that a deal will be concluded, as Pepe ends an underwhelming four years in the English capital.
Arsenal smashed their transfer record to bring the Ivorian to the club in 2019, but he failed to rediscover the prolific form that earned him rave reviews in Ligue 1, netting 27 times and setting up 21 more goals in 112 matches.
The emergence of Bukayo Saka saw Pepe demoted to a peripheral role under Arteta, who sanctioned his loan exit to Nice for the 2022-23 season, where the attacker came up with eight goals in 28 games.
Injuries plagued Pepe's time at the Allianz Riviera, though, and he has not made a single matchday squad for Arteta in the 2023-24 season as the club try to cut him from the ranks.
The former Lille man is only under contract with Arsenal until 2024 and will be able to leave for free next year if no sale is agreed before then - a scenario which the Gunners are desperate to avoid.
Arsenal waved goodbye to seven players on permanent deals this summer, as Granit Xhaka, Folarin Balogun, Matt Turner, Pablo Mari, Rob Holding, Ainsley Maitland-Niles and Auston Trusty left for pastures new.
Meanwhile, Kieran Tierney, Marquinhos, Albert Sambi Lokonga, Alex Runarsson, Nuno Tavares and Arthur Okonkwo have all left the club on loan, while Cedric Soares could still depart in the coming days after being frozen out of the squad.
Arsenal's first game back after the international break sees the Gunners square off against Everton at Goodison Park on September 17.