The England international, who failed to make the cut for the ongoing World Cup, is out of contract at the Emirates this summer although he has been offered a new deal.
According to the paper, the 26-year-old, who made his debut for the Gunners at 16 back in September 2008, is pondering whether to sign it or not with new manager Unai Emery not prepared to offer him a guaranteed place in the team.
Palace are interested in taking Wilshere across London with their boss, Roy Hodgson, having previously worked with the player during his time in charge of England. Wilshere's last cap came under Hodgson at Euro 2016.
West Ham, Wolves and Everton have also been linked with moves for Wilshere, whose services look set to be in demand should he opt to leave.
Serie A clubs AC Milan, Sampdoria and Juventus are also reportedly interested.