Januzaj is moving on from Spanish La Liga side Real Sociedad as a free agent and is not short of offers.
The former Manchester United winger has attracted the eyes of West Ham boss David Moyes, but it is unclear if the Hammers have sent a contract offer to him.
Napoli have taken that step though and, according to Italy-based journalist Ekrem Konur, Napoli have offered Januzaj a three-year deal.
The Serie A side want to tempt the winger to continue his career in Italy next season.
He made 32 appearances in La Liga for Real Sociedad last term, scoring three times, and also turned out for the club in their Europa League run.
Januzaj has never played his club football in Serie A and could be attracted to the idea of a move to Italy.
He made 63 appearances for Manchester United during his stint at Old Trafford and turned out under Moyes at the club.