The Manchester United goalkeeper is on a season-long loan with the Turks, though no permanent option was included in the deal.
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Onana is being celebrated this week by Trabzonspor fans after saving three penalties in their Turkish Cup quarterfinal shootout win against Samsunspor.
Doğan, speaking to NTV Spor, has now stated: "Onana's career is already established.
"We like him. I've said this before. He has a certain plan for his career. If the conditions are right, we want him to stay. The final decision will be Onana's."