The Daily Mirror says Napoli are weighing up a £13.1m bid for Wilshere.
The Gunners midfielder has recently returned to the first team fold in north London.
However, the Serie A outfit want to offer Wilshere first team football to boost his World Cup chances.
Wilshere is also a confirmed target for West Ham boss David Moyes.
Moyes admitted: “Jack would be someone who we'd have to look at if he was available.
“I want to make sure we're looking at players who've got time and who can be at the club for a long period, not just in for a short period."