The Rossoneri are in final stages of completing the deal for the young attacker thought to be in the region of £31 million, with super-agent Jorge Mendes pulling the strings from behind.
The report claims that Milan have accelerated the negotiations for Leao, while working in parallel with the operation for Angel Correa.
The deal is as good as done and only a few details are left to sort it out between the two clubs.
The news will come as a huge blow for Everton who are reportedly keen on signing the Lille attacker.
French newspaper L’Equipe (translated by GFFN) claimed earlier this month that Everton had already submitted a €37.5m (£33.6m) offer for the forward.
In fact, the Merseyside club were the highest bidder in the race for his signature.
Everton either have pulled the plug on the deal or they have contemplated way too long allowing Milan to swoop in and hijack the move.
The Toffes are concentrating on other targets though with Marco Silva keen to sign Wilfried Zaha from Crystal Palace and Moise Kean from Juventus.