Demir, 19, joined Galatasaray this summer and has made two appearances for the Turkish giants in the season thus far.
The reason for his few appearances is that he has struggled with injuries, with a thigh muscle strain currently keeping him out.
Despite him not yet starting for Galatasaray in the league and moving to Istanbul just in the summer, he is being linked with a move away already.
Both Manchester United and Liverpool have added his name to their transfer list, according to German outlet RUHR24.
Apart from Manchester United and Liverpool, German duo Borussia Dortmund and Bayer Leverkusen are also said to be interested in the winger.
The 19-year-old was wanted by Borussia Dortmund even last summer but he ended up leaving Rapid Vienna for Barcelona on loan, making nine appearances for the Catalan side.
Demir has a contract until the summer of 2026 with Galatasaray and it remains to be seen if the Turkish giants are willing to let him go next year.
