According to Turkish outlet Fotomac, the 37-year-old is on the Istanbul club’s radar as they seek to fix their ongoing attacking woes.
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Fenerbahce are already well behind Galatasaray, sitting six points their arch-rivals in the race for the 2025-26 Super Lig title.
Manager Domenico Tedesco is understood to have told club president Sadettin Saran he wants a proven goalscorer and has sounded Benzema out for a January move.
The 2022 Ballon d’Or winner and former Real Madrid superstar is still under contract with Saudi club Al-Ittihad until next summer.