The former Gunners U18 coach is available after leaving his assistant coaching post at Norwich City last season.
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Wilshere left Arsenal for Carrow Road a year ago and was offered the U21 job by the Gunners after Mehmet Ali's departure for Brentford.
BBC Sport says Wilshere held talks with Arsenal chiefs about a return, but after consideration chose to decline the offer.
Wilshere ended last season as caretaker manager at Norwich and is expected to pursure a senior post for his next move.
Max Porter and fellow coach Ken Gillard have been in caretaker charge of Arsenal U21s this preseason. Porter is now favourite for the permanent post.