Piatek, 23, has been one of the breakout stars in Europe in his debut Serie A season, after netting 19 goals in 21 appearances for Genoa.
His form attracted the interest of LaLiga giants Real Madrid and FC Barcelona but according to The Guardian, Milan have won the race for his services.
It marks another step in his meteoric rise, having only arrived in the Italian top-flight in June in a €4 million switch from Cracovia.
Piatek, who will sign a five-year deal, will replace Higuaín at Milan with the Argentinian striker set to complete a move to Chelsea this week.
Higuaín has been left out of Milan’s game against Genoa on Monday with his manager saying that the 31-year-old “was not mentally ready” to play the game.
Gennaro Gattuso added that Higuaín had produced “one of the worst training sessions” the manager had seen and had decided to leave him out after that.
"All this talk is not easy for him," he added.
Higuain's arrival will also pave the way for Alvaro Morata to complete his expected LaLiga return to Atletico Madrid until the end of the season.