The 35 year-old man, who was with a group of Inter Milan fans, was taken to hospital in the northern city on Wednesday evening but died in hospital.
The clashes took place not far from Milan's San Siro stadium before a Serie A game.
Four Napoli fans were hurt and one of them was stabbed, police said.
The clashes took place in a street a few minutes' walk from the San Siro stadium, reports said.
Around 10 minivans carrying Napoli supporters arrived around an hour before the match and evaded police security checks near the stadium.
The vans were surrounded by more than 100 people armed with sticks. Two people appeared to be knocked over, and one of them taken to hospital with critical injuries.
The man who died has been described as an "ultra" from the town of Varese to the north of Milan who was given a ban six years ago for his part in clashes at another Inter match. Italian reports named the victim as Daniele Belardinelli, aged 35.
It is unclear how he was knocked down.
Police told reporters he had not been hit by a minivan from Naples and that they were studying CCTV footage of a dark SUV at the scene.
At a news conference on Thursday, city police chief Marcello Cardona said he would seek to ban Inter fans travelling to away matches.
He was also seeking a five-game closure of a section of the San Siro stadium known as the Curve that hosts the Inter ultras.
Three Inter fans were arrested over the clashes that took place ahead of the game, he said.
Tensions were high inside the stadium as well, where Napoli boss Carlo Ancelotti accused Inter Milan fans of repeated racist chanting.
Senegal defender Kalidou Koulibaly, who was sent off, had been targeted to such an extent, said Ancelotti, that the coach had asked three times for the match to be suspended.
Milan Mayor Giuseppe Sala apologised for the fans' abuse of Koulibaly, saying, "Those boos against Koulibaly were a disgrace."
Napoli and Inter are second and third respectively in Serie A, some way behind Juventus.
Interior Minister Matteo Salvini tweeted that in 2018 "you cannot die for a game of football".
He promised to convene a meeting of supporters' leaders from the two top flights of Italian football to ensure that "stadiums and the areas surrounding them become a place of fun not violence".