The Spaniard insists he is looking forward to working with new manager David Moyes and claims he still needs to improve to be considered as one of the world's best goalkeepers
The 71-year-old Scot called an end to his career last season after 26 years at Old Trafford.
David Moyes will officially take over from Sir Alex on July 1 and the 22-year-old is looking forward to a change in manager at United.
"He [Sir Alex] is the best coach history," De Gea told reporters. "I remember that the best advice he gave me is simply to give it everything. He's asked me to continue with this mentality.
"I have wanted to start a new phase in Manchester and hopefully it will go well."
After arriving at United from Atletico Madrid in June 2011, De Gea came under criticism following some of his early performances.
De Gea, who is currently with Spain's Under-21 squad for its European Championship campaign in Israel, believes he has developed during his time at Old Trafford but insists he still has much to learn to be considered one of the best goalkeepers in the world.
"I've grown a lot," he continued. "I think I still must grow to be among the best."