The Nigeria striker is on-loan at Gala from Napoli and can be signed permanently for a fee of €75m courtesy of a clause in his Azzurri deal.
Osimhen is expected to leave Gala and Napoli for a bigger competition, but the Turkish giants remain convinced they can keep hold of the centre-forward.
Galatasaray's management team has stated that they are willing to meet all the necessary financial and sporting conditions for Osimhen to stay.
"We will make a decision soon. It will be decided what is best for me. Galatasaray will always be in my heart...," declared the forward yesterday, who has scored 21 goals this season in Turkey.
Club vice president İbrahim Hatipoğlu spoke to AS and clarified that Osimhen's words should not be interpreted as a final farewell.
"His words don't mean he's leaving. He said that even if he left in the worst-case scenario, Galatasaray would always be in his heart. But he didn't say he's leaving. We want him to stay with us," Hatipoğlu insisted.