De Gea moved to United yesterday.
De Gea, who has long been linked with a move to United, said: "There have been many rumours these last few months about if I had gone there or not, but the truth is that last week there was contact from Manchester United to Atletico Madrid saying that they would like me to visit their facilities and present an offer to me.
"I decided to go with my family and study the offer and see everything and that's when I decided.
"They have very good facilities. It's one of the best clubs in the world and I'm going there to continue growing as a player and as a person, and for that it's a perfect club.
"Atletico made an effort (to keep me), but my family and I have taken this decision."
He added: "I want to thank everybody who has had faith in me and the fans who have always supported me since I was young. I will always carry Atleti in my heart."