The Ivory Coast international has been told that his Gunners career is dead in the water, and a departure is seen as inevitable after he was frozen out of Mikel Arteta's squad.
With the Saudi Arabia window closing at 10pm on Thursday, a move to the Middle East is not expected to come to fruition for Pepe, who is instead being tipped to head to the Super Lig before the Turkish window shuts on September 15.
A transfer to Besiktas was originally seen as his most likely escape route from North London, and the 28-year-old had apparently already passed a medical with Senol Gunes's team.
However, Pepe's move to Besiktas is believed to have collapsed after Arsenal wanted to revise the terms of the deal, and Trabzonspor have since jumped to the head of the queue.
Now, transfer guru Fabrizio Romano reports that the former Lille man has given the green light to Trabzonspor's contract offer, and there is hope that a full agreement will be sealed imminently.
Romano adds that Pepe turned his nose up to the chance to move to Saudi Arabia, as the Arsenal man only had eyes for a move to Trabzonspor, who are now working to agree a fee with the Gunners.
Arsenal paid a then-record £72m to bring Pepe to the Emirates in 2019, but with the Ivorian's contract expiring at the end of the season, the Gunners will have to accept a significant loss to get him off the books.
A separate update from RMC Sport journalist Fabrice Hawkins claims that Arsenal want to recoup between €4m (£3.4m) and €5m (£4.3m) to let Pepe depart, although the Gunners could still terminate his contract in the coming days.
The winger has consistently flattered to deceive since becoming Arsenal's most expensive player in 2019, although he was stripped of that title by £105m man Declan Rice in July.
Pepe has come up with 27 goals and 21 assists in 112 games for Arteta's side, winning one FA Cup in his debut campaign during the 2019-20 season, but he has not played a competitive game for Arsenal since May 2022.
Nice offered him the chance to reignite his career in Ligue 1 during the 2022-23 season, and while he managed eight goals in 28 games during a respectable loan spell, he was plagued by injuries.
With Bukayo Saka, Fabio Vieira and Reiss Nelson all capable of playing on the right-hand side, Pepe was not given the chance to get back into Arteta's good books in pre-season and will soon end a disappointing four-year spell in the Premier League.
The 28-year-old will become the eighth player to leave the Emirates permanently this summer, following Ainsley Maitland-Niles, Folarin Balogun, Granit Xhaka, Matt Turner, Pablo Mari, Rob Holding and Auston Trusty out of the door.
Meanwhile, Alex Runarsson, Marquinhos, Kieran Tierney, Nuno Tavares, Arthur Okonkwo and Albert Sambi Lokonga have all left on loan, with Tavares's move to Nottingham Forest also including an option to buy.