The report states that both the Gunners and Red Devils are keen to bring the Turkish international to the Premier League.
Taksport go onto states that the La Liga champions Barcelona are also in the race to sign the 21-year-old playmaker.
The Turkish international starred in the Crescent-Stars 1-0 victory over Iceland that booked Turkey a place in Euro 2016.
Çalhanoğlu started against Iceland at the Konya Torku Arena and is expected to be a key player for Turkey in the European Championships this summer in France.
The Bayer Leverkusen midfielder is one of the most sought after players in Germany and has a brilliant record in Europe playing a direct role in 12 (6 goals, 6 assists) in his 14 European games.
Çalhanoğlu has already scored six goals and provided two assists in all competitions this season. The Bayer play-maker star also has a phenomenal free-kick scoring record – 22 in the Bundesliga alone.