L'Equipe says Ederson, 31, is the subject of a €3m offer from Galatasaray this week.
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The Brazil international is also a big money target for several Saudi Pro League clubs this summer.
However, before the end of last season, Ederson insisted: "My head is here, it stays with City. My future is here.
"It gives me everything to go back to the Premier League again and try to win the Champions League as well. My mind is staying in the club."