AC Milan spent in excess of £170million on new recruits in the summer but have endured a mixed start to the season as they sit sixth in Serie A.
And Italian media outlet Gazzetta dello Sport claim manager Montella, who took charge in June 2016, is already under pressure after just 10 games of the new season.
Montella is said to have the next two games to save his job against rivals Roma and Inter Milan.
Milan’s wealthy owners have apparently made contact with Chelsea boss Conte and Bayern Munich manager Carlo Ancelotti.
Chelsea manager Antonio Conte shows off impressive footwork
AC Milan contact Chelsea manager Antonio Conte
Ancelotti spent five years at Milan during his playing career and another eight years as their manager, winning 15 honours in the process.
While Conte enjoyed a successful three years in charge of Juventus before spells at Italy and now Chelsea, winning Serie A in each of his seasons in charge.
Conte, who won the Premier League in his first season at Chelsea, this week admitted he plans to return Italy at some point in the future.
"It is not in my mind to stay abroad for a long time,” he said. "Italy is my country. I will return, I don't know when but that is the aim."
And asked further about his future and other countries that he may enjoy working in, including China and Germany, Conte said: "China? No, absolutely not.
“Bundesliga? It's always difficult to predict the future. Let's just say that a coach is a temporary worker.
"It depends on the results, but there's desire on my part to try to do the best I can, finishing one project and then finding the right one to continue. I miss Italy, that's a fact.
"Still, this experience is helping me so much. I would also like to change role, maybe become a sporting director in the future."
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