He is thought to be in Istanbul to complete a medical ahead of signing a one-year deal, according to transfer guru Fabrizio Romano.
He was filmed donning Trabzonspor's colours on the flight to Turkey while saying: "I came here to become a champion and achieve success."
Pepe has one year left on his current Arsenal deal and has not been included in any of manager Mikel Arteta's squads this season.
He spent all of the last campaign out on loan at French side Nice and Arsenal have since been looking at suitors for his signature over the summer.
It brings an end to a disappointing four-year spell in North London after he became Arsenal's record signing for £72 million in 2019.
Pepe scored 27 goals in 112 appearances across all competitions for the club.
