The Gazetta dello Sport has suggested that the southern Italian side considering a bid for the German World Cup winner.
Carlo Ancelotti recently revealed that Khedira has been told he is allowed to leave Real Madrid with just one year left on his contract.
His future has been in doubt for most of the summer, and the Gunners are believed to have been monitoring his situation for some time.
Napoli have a recent history of signing Real Madrid players, having signed Raul Albiol, Gonzalo Higuain and Jose Callejon last summer.
With Marouane Fellaini having suffered an injury, Napoli are thought to have turned their attentions to the German as a defensive midfield option.
Khedira suffered serious knee ligament damage last season but returned to start Real Madrid’s victory in the Champions League final, and he played a key role in Germany’s World Cup win in Brazil this summer.