The Partenopei have been linked with the 26-year-old for some time, but according to the Corriere dello Sport they are stepping up their efforts.
Callejon is thought to have already given his all-clear to the move, although there is a €2.5m (£2.1m) gap between the current offer and Real Madrid’s €10m (£8.5m) price-tag.
There is also an issue to work out with regards to the image rights, but Italian media are optimistic.
Napoli are also on the verge of signing Santos goalkeeper Rafael for €5m (£4.2m).
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- Sports & Recreation
- Corriere dello Sport