The 18-year-old striker has accounted for nine goals and one assist in 15 appearances in the Sper Lig this season.
Turkish transfer expert Ekrem Konur is reporting: "Borussia Dortmund, Atletico Madrid, Tottenham , Manchester City and Arsenal scouts watched many matches of Beikta's 18-year-old Turkish striker Semih Klsoy this season."
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Kilisoy has a contract with Besiktas until the summer of 2028.
Besiktas is currently in second place in the Turkish league, behind league leaders Galatasaray.