The Spain international came close to a move to the Bernabeu in the summer before the proposed deal fell through, with De Gea subsequently signing a new deal at Old Trafford.
Madrid were hit with a transfer ban last week, which bans them from signing players in the summer and next January, leading to reports they could move for De Gea in January before the suspension begins.
However, De Gea is adamant he no longer has any desire to leave the club and says he is focused on emulating the achievements of Schmeichel and Edwin van der Sar.
"I'm happy, I'm enjoying it and I believe in the team," De Gea told Sky Sports.
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"I train every day to be the best goalkeeper in the world, that is my aim. I'm not sure who the best is now, but it's what I want. Also, to win trophies for Manchester United."
De Gea also recalled the last time he faced Liverpool back in September, when he received a rapturous reception from fans on his return to the side after signing a new contract, following interest from Real Madrid.
"I cannot describe how I felt in that moment," he added.
"In another country, at another club, the fans would not be the same as at Manchester United. They are some of the best in the world, it was unbelievable."