Claudio Ranieri has been placed in charge until the end of the season, but iconic figure Francesco Totti is said to be pushing for the club to bring in Gattuso ahead of the 2019-20 campaign.
Gattuso could also be out of work in the coming weeks as reports in Italy suggest Milan are after a new coach to take them forward following rumoured tension behind the scenes between the 41-year-old and sporting director Leonardo.
However, Milan's sporting strategy and development director Paolo Maldini praised Gattuso this week for showing "great character" in the face of off-field adversity, with the Italian giants being probed by UEFA for possible Financial Fair Play irregularities, and the ex-midfielder has suggested he intends to stay put at San Siro.
"I really enjoyed the words of Maldini yesterday," he said at Friday's news conference. "I still have a two-year contract. My goal is to return to the Champions League, the rest is just gossip."
Milan have taken just one point from the last 12 on offer in Serie A, allowing the chasing pack to close the gap on them in the race for a top-four finish.
Gattuso's side occupy the final Champions League qualifying spot, but they are level on points with Atalanta and just one better off than Roma with seven games to go.
An intense training at Milanello in the presence of Ivan Gazidis, Leonardo and Paolo Maldini— AC Milan (@acmilan) April 9, 2019
Secondo allenamento della settimana per i Rossoneri, sotto gli occhi di Ivan Gazidis, Leonardo e Maldini pic.twitter.com/jECrIuxvFp
Ahead of Saturday's home match against seventh-placed Lazio, who will draw level on points with Milan if they win, Gattuso has called on his players to match their display from last week's 2-1 loss at leaders Juventus.
"Lazio are a very physical team, in this aspect they are like Juventus,” he added. "They always fill the penalty box well. Tomorrow, it won’t be enough to just play well, we will need great desire. We will have to play in the best possible way.
"I’d be happy to see us replicate the performance against Juventus, obviously without certain mistakes. I want to see the same mentality as a week ago. It has been a good week, we trained well."
Milan will still be without goalkeeper Gianluigi Donnarumma and influential midfielder Lucas Paqueta for this weekend's clash because of a muscular injury and sprained ankle respectively, both sustained in the first half of the 1-1 draw with Udinese on April 2.
Providing an update on the duo's fitness, Gattuso said: "Gigio should be with us again next week. Paqueta is improving."