Wilshere has been with the Gunners since 2001 but the long-serving player will become a free agent if he does not pen a new contract by the end of June.
The England international has been linked with a wide range of different clubs but according to L'Equipe, PSG have been given the chance to sign the player.
While the 26-year-old has missed out on a place in England's squad for the World Cup, the playmaker remains an attractive option for many top clubs in Europe and PSG may jump at the opportunity to sign a player of Wilshere's calibre without a transfer fee being involved.
Despite concerns over his fitness, Wilshere has made 67 appearances in all competitions since the start of the 2016-17 campaign.
Arsenal have allegedly offered the academy graduate a new three-year contract.