The 24-year-old Wilshere, who was tipped to become a midfield superstar a couple of years ago, has been hampered by injuries in recent seasons and was not selected in England's squad for their upcoming World Cup qualifier against Slovakia on Sunday.
He made only made three appearances for Arsenal last season after fracturing his left fibula.
The Gunners have an abundance of players that can play in midfield, including new signing Granit Xhaka, and Wilshere will struggle for game time even when he is fit.
With the transfer deadline looming, Wilshere has less than two days to find a club that is willing to make him an offer.
Fellow Premier League clubs Middlesbrough and West Ham, as well as La Liga side Valencia, have been rumoured as possible destinations.