The 41-year-old is set to sign a six-month contract with the Serie A club, who will have the option to extend the deal by a further year.
Ancelotti was sacked on Tuesday night, just hours after leading Napoli into the last 16 of the Champions League with a 4-0 win over Genk.
However, that was their first win in 10 matches and Ancelotti’s relationship with club president Aurelio De Laurentiis had become strained this season.
Gattuso, who played under Ancelotti at AC Milan, is set to take Napoli training on Wednesday.
Following Ancelotti’s departure from San Paolo, Everton are set to begin talks over making him their new manager following the departure of Marco Silva last week.
Sky in Italy reports that Arsenal are also interested in the 60-year-old as they search for Unai Emery’s replacement.