The Spaniard joined the Red Devils two years ago from Atletico Madrid and initially struggled to adapt to the Premier League, but last season was voted in the PFA Team of the Year.
De Gea told The Telegraph: "When Edwin van der Sar said he was leaving Manchester United I wanted to be the man to replace him. I was given that chance and I am not going to let it go.
"Being the United goalkeeper is one of the most difficult jobs in football when you look at the size of the club and the goalkeepers who have been here before.
"There is a lot of pressure on you, but I have worked hard to cope with that and I would love if, one day, the fans speak of me in the same way they speak of Edwin and Peter Schmeichel."
De Gea made 28 appearances in the Premier League for United last season.