Former manager Vincenzo Montella was sacked on Monday after a disappointing beginning to the season for the Serie A giants, with Gattuso promoted from his position with the Primavera to take over the first team.
Despite this, Milan's "ultimate dream" remains the capture of Conte, according to Fabrizio Romano of Calciomercato.
It's suggested Milan "would've let Montella go last summer if Conte was free to join them," and while there is faith in Gattuso to do well "this won't stop Milan from dreaming about Conte."
Additionally, it's also reported director Massimiliano Mirabelli was in London earlier in the month and held talks with the 48-year-old's representatives in which it was made clear he was their primary target to take over at the end of the campaign.
Gazzetta dello Sport (h/t Football Italia) has also reported that the former Italy manager is at the top of Milan's shortlist along with the team's former coach Carlo Ancelotti. The latter is out of work after he was axed by Bayern Munich earlier in the season.
The decision to give Gattuso the top job at Milan is a fascinating one, although Bleacher Report's Adam Digby doesn't view the 39-year-old as a long-term appointment:
He will at least be a popular figure with the San Siro crowd, having enjoyed a tremendous playing career with the Rossoneri.
Gattuso was the engine in the middle of the Milan midfield at his best, complementing the likes of Andrea Pirlo and Kaka in some fantastic football teams. In a 13-year spell with the club he won two Serie A titles and the UEFA Champions League on a couple of occasions.
He's been in management since 2013, taking over as a player-manager at Sion in 2013. As we can see courtesy of Scouted Football, his coaching path has not been an orthodox one:
It's why Milan might be looking at different names for the summer. And having spent such an incredible amount on new players ahead of the current campaign, they'd surely be willing to splash out significant funds on a manager.
Conte, you'd have to say, would be worth a massive outlay, as he's proved himself as one of the best coaches on the planet.
At Juventus he won three consecutive Serie A titles, before taking over the Azzurri position. He was then able to take an underwhelming Italy squad to the UEFA European Championship quarter-final, where they lost on penalties to Germany, and then steered Chelsea to Premier League glory in his debut campaign.
According to Romano, the Italian is happy with life at Stamford Bridge and a "potential deal remains very complicated indeed." Still, the chance to return to his homeland and possibly restore a giant like Milan as one of the biggest forces in European football would surely be an appealing project for the charismatic coach.