Wilshere has started in each of Arsenal's last two fixtures in the Premier League, but the 25-year-old still has just under six months remaining on his contract.
Manager Arsene Wenger has indicated that he wants Wilshere to prove his fitness before offering the playmaker a new deal but according to Calcioline, Napoli are ready to make their move at the start of 2018.
Should Wilshere reach January 1 without penning fresh terms, he will be able to speak to foreign clubs regarding a free transfer in the summer, but it has been suggested that Napoli would like to sign the England international midway through the campaign in order to boost their hopes of winning the Serie A title.
The report claims that Napoli would be prepared to pay in the region of £13.2m for the long-serving Gunner, who is keen to earn a place in Gareth Southgate's squad for the World Cup.
One potential stumbling block could be Wilshere being ineligible to represent Napoli in the Europa League, having already featured for Arsenal.