Napoli announced Ancelotti's departure in a short statement posted on social media just hours later. "The friendship and mutual respect between the club, president Aurelio De Laurentiis, and Carlo Ancelotti remain intact".
Former Rangers midfielder Gattuso joins Napoli at a hard time despite their arrival in Monday's draw for the knockout stages of Europe's top competition.
He has been linked to the vacant manager jobs at Arsenal, Everton and Major League Soccer expansion club Inter Miami.
The World Cup-winning former Ger has had a checkered managerial career and left his last role as AC Milan boss at the end of last season.
"I hope that they will be back at the top as they were for years until this bad moment".
Mills acknowledged that Ancelotti had an exceptional CV and ticked a lot of boxes and suggested he is someone the Toffees should consider once they secure their Premier League status.
"It's embarrassing to see Napoli where we now are in the table".
A number of fines were dished out to the players but Ancelotti defended his men, a move that De Laurentiis never forgave. "In all these years he has always been near me and helped me in every situation".
The Italian squad sits in seventh place in Serie A. Napoli now lie in seventh place in Serie A, eight points adrift of Cagliari who now occupy the last Champions League berth.