Toffees boss Carlo Ancelotti worked with Milik during his time at the Stadio San Paolo, scoring 20 goals in 47 appearances during the Italian's only full season with the Serie A outfit.
Arena Napoli now claim that Ancelotti is exploring the possibility of bringing Milik to Merseyside, with Everton having already made preliminary contact with the 26-year-old.
Everton are thought to be lining up a bid of as much as €35m (£26.5m) for the Poland international, but Napoli are holding out for an offer closer to €50m (£44.1m).
The report also claims that Everton have tabled a contract worth €4m (£3.5m) per year, which is said to be the most attractive offer so far.
Milik, whose contract in Naples expires next summer, has scored 12 goals in 22 appearances this season, including nine in just 16 Serie A outings. body check tags ::