Romero, 20, is yet to play for his national side Argentina and only joined Genoa from Begrano last summer.
But the defender has his heart set on a move to one of the biggest clubs in Europe.
There are claims Juventus have already got an agreement in place to sign the promising youngster.
But Romero has opened the door on a potential move to Old Trafford by insisting he is in the dark over his future.
“I’ve always wanted to become a professional football player and I’ve made it happen,” he told Corriere dello Sport.
“Now I dream of playing in the Champions League for one of the four most important clubs in the world: Juventus, Real Madrid, Barcelona or Manchester United.
“After that, I want to be part of the Argentine national team, even if I know it’s difficult because there’s Otamendi, one of the best in the world.
“Have Juve already secured me? I don’t know. I don’t deal with the market and my first thought is Genoa.
“There’s still so many matches remaining and I want to keep playing like I am. After that, I’ll talk to the directors and my agent.”
Romero made his Genoa debut against Juventus back in October and came up against former United star Cristiano Ronaldo.
He was booked in that match and, in the next game, scored and was sent off in a 2-2 draw with Udinese.
Booked seven times in 16 appearances already this season, the combative Romero insists he can handle himself in Serie A.
Asked about facing Ronaldo on his debut, Romero said: “So many people have asked me this question but I have a quiet character and, when I play, I don’t think about who I have in front of me.
“Ronaldo‘s one of the best in the world, [Paulo] Dybala is also very good and Juventus is a great team, but I’m only focused on giving my best.
“[Torino striker Andrea] Belotti is the one who has bothered me the most: he moves a lot and is strong physically.”