The Napoli-owned striker led Gala to a Super Lig and Cup Double last season and can be signed permanently by the Turkish giants for €75m.
Discussing their summer work so far, Okan said: "The negotiations for Osimhen are going very well. We're now close to completion, we expect it to be finalized as soon as possible.
"The player's desire is there, and Napoli is also showing a constructive attitude. We hope it can be concluded soon. Our entire transfer window depended on Osimhen, and there was a very significant budget involved. Only after finalising the deal will we be able to understand where and how to spend the rest of our budget."
Positive Zaniolo signs
Okan also welcomed the arrival of Nicolo Zaniolo after last season's loan with Fiorentina.
He added, "Zaniolo is a very talented player. We want to monitor him closely during this preseason period. He's working well, everything is going in the right direction.
"In friendlies, he's shown positive signs in terms of discipline and understanding of the game. He's an important asset for us."