The Eagles nearly landed the England international last summer, however Eddie Howe’s Bournemouth secured the midfielder’s signature on transfer deadline day last August.
Newly appointed Palace boss Frank de Boer is also an admirer of Wilshere and his playing style and is happy for Palace to go for him again.
A source told SunSport: “Crystal Palace have made an approach and want Jack for the season.
“Jack is out of contract at the end of the season and Arsenal are offering him a new one. They will only loan him out if he signs a new contract – otherwise they want to sell him or keep him at the Emirates.
The 25-year-old is close to returning to full training after fracturing his left leg in a collision with Tottenham’s Harry Kane in April.
Wilshere has struggled with injuries throughout the majority of his career, however, he played 27 times for Bournemouth in the Premier League last season as the Cherries secured a ninth-place finish.
Should the talented midfielder move to Selhurst Park, it will be his third loan spell away from Arsenal having played for Bolton Wanderers in 2010.
Other Arsenal players could also be departing the club this summer. Keeper Wojciech Szczesny is currently in Turin having a medical with Juventus, while Lucas Perez, Carl Jenkinson, Mathieu Debuchy and Kieran Gibbs have all not travelled on their pre-season tour and will be allowed to leave the club.