The 19-year-old has been linked with a move to the Spanish club following an impressive season in the Bundesliga with the Royal Blues.
"Real Madrid are one of the world's great clubs, it makes me proud that they like my style of play," he told Marca. "It would be a dream to play at Real Madrid.
"I've always said that I would like to play in Spain. Every weekend I try to see the matches involving Madrid and Barcelona. I never miss the Clasicos.
"Right now I'm with Schalke and I have a contract until 2016. I'm having a very good season and I'm doing very well, I've scored eight goals in the Bundesliga and I feel very happy."
Draxler made his Bundesliga debut for Schalke in January 2011.