The Argentina international has been urged to find himself a new club after falling out with manager Ruben Amorim at the end of last season.
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Garnacho has been linked with Chelsea and Aston Villa in recent weeks.
However, Milliyet says Fener are now making their interest known, where coach Jose Mourinho is fully backing the move.
Fener are prepared to "push themselves to the limit" to bring Garnacho to Istanbul, with United rating the player at £50-70m.