United watched the attacker in action for Gornik Zabrze only in November before he agreed to join Bayer.
"My dream is to play for this club. I'm a Reds fan, so I would love to one day play in the team at Old Trafford. So far I am a footballer of Bayer Leverkusen and Manchester is very far away," said Milik.
Asked about the prospect of himself and Borussia Dortmund striker Robert Lewandowski playing together at United, the youngster added: "It would be great to play on the side of Robert, especially at Manchester United.
"First, I have to give my best in training and in matches to prove that I am up to standard for Bayer Leverkusen."