The 26-year-old has been with the Gunners since the age of nine, joining his boyhood club in 2001 and going on to make almost 200 first-team appearances for the North Londoners.
However, Wilshere's contract is due to expire at the end of this month and talks over an extension have so far failed to see the two parties reach an agreement.
According to The Mirror, Wilshere was desperate to stay at the Emirates Stadium and had agreed a deal but, after talks with new boss Emery this week, he has been told that he could end up being a fringe player rather than a regular starter.
On Thursday, the England international suggested that he would soon be ending his 17-year association with the club, posting a quote on his Instagram story which read: "There comes a time when you have to choose between turning the page and closing the book."
Wolverhampton Wanderers, West Ham United and Everton are among the clubs that have reportedly offered Wilshere deals but there is also believed to be interest from abroad with Sampdoria, AC Milan and Juventus said to be monitoring the situation.