Per the source, the Gunners want to loan Wilshere out in order for him to lock down a regular first-team role and regain form after spending most of last season out injured. The report also claims that Besiktas are interested.
Wilshere has been closely linked with rival Premier League moves but the North London based outfit reportedly want the England international to move abroad and gain experience in a league overseas.
Besiktas would be able to offer Wilshere first-team and Champions League football this season if he were to make the move.
The Black Eagles won the league title last season and will compete in the Champions League Group B against Benfica, Napoli and Dynamo Kiev.
Wilshere would join former Arsenal youngster Oguzhan Ozyakup at Besiktas if the transfer goes ahead.
Besiktas would only be prepared to loan Wilshere as they are currently under the watch of FFP and would not be able to offer a transfer fee for a permanent role at the club as a result.