The 25-year-old midfielder is one of seven players due to see their current terms expire at the end of the campaign, meaning that he is open to leave as a free agent.
Wilshere has seen his Emirates Stadium career stagnate in the past few years, leading to a season-long loan spell at Bournemouth last time out which - according to The Sun - has left him wanting a permanent change.
Wenger claimed earlier this week that contract talks with Wilshere have yet to begin, but the same report suggests that the Englishman was offered the chance to extend his deal in the summer, which he knocked back.
A source close to the player told the publication: "Arsenal didn't want to loan him out again unless he signed a new contract — but he didn't want to sign one.
"If Jack gets back into the Arsenal team on a regular basis and feels he is seriously part of the club's plans, there is still a chance he might stay.
"However, as things stand, he is still of the view that he will be leaving at the end of the season. But he will be giving it his all for the club and will remain professional."