The Goodison Park club are embarking upon the hunt for a new manager after taking the decision to relieve Rafael Benitez of his duties on Sunday.
A host of managers have already been linked with the post at the Premier League side, including Belgium coach Roberto Martinez and Derby County boss Wayne Rooney.
Another manager is also under consideration at Everton in the shape of former Napoli coach Gattuso.
Gattuso held talks over taking over at Tottenham Hotspur last summer before the north London side eventually appointed Nuno Espirto Santo.
He is available and Everton are claimed to have reached out to his agent Jorge Mendes to kick off contact.
Gattuso, 44, had a stint in charge of AC Milan before he took over at Napoli, where he then led the southern Italian side to the Coppa Italia in 2020.
The Italian has experience of British football gained as a player at Rangers in the 1990s and could be keen to answer the call from Goodison Park.