The 29-year-old is reunited with Toffees boss Carlo Ancelotti, with whom he worked at the Serie A club.
Everton are also close to reuniting Ancelotti with Colombian star James Rodriguez for a third time after spells together at Real Madrid and Bayern Munich, while a fee has reportedly been agreed with Watford for Abdoulaye Doucoure.
Allan, who made more than 200 appearances in five years at Napoli, admitted he could not turn down the opportunity to work with Ancelotti again.
It is a club with a rich history in the Premier League, has real ambition and then there is Professor Ancelotti. He has done everything possible to bring me here,” Allan told Everton TV.
“It is the size of the club and the name of the coach which means you don’t think twice about coming to Everton.
“I hope, like I have done in my entire career, I contribute with my performances together with my team-mates and that I put in some great games, great performances and win important things.”