The 21-year-old arrived in the summer with a big reputation following a series of impressive displays between the sticks for Atletico Madrid.
A big transfer fee saw the pressure mount on the Spaniard and, after a series of shaky performances, many questioned whether he is the right man for United in the long-term.
De Gea, however, maintains his self-belief in his abilities and is confident he can prove to be a success at Old Trafford.
"I have every intention of spending many years at Manchester United,
" said the Spaniard.
"I want to go on to become a great United keeper and to earn and deserve respect. I want to help United make history."
The shot-stopper's frail physique was part of the problem with his start to life in English football, with a number of teams targeting the goalkeeper as a means of scoring.
De Gea admits that it has taken time for him to adjust to the rigours of the Premier League, but always knew he was going to face such challenges following his move.
"I think it is fair to say La Liga
and the Premier League are the two best leagues in the world," he added.
"But it's well known the style of play here is rather more physical and, as a goalkeeper, you have to be right on it. You have to be attentive all through the match.
"The players coming at you can shoot from any angle and from distance. You need to be aware you might get a shot from distance at any time.
"Having said that, there's a lot of crosses from the sides, a lot of high balls and aerial play. So you have to be able to get in there and block, interfere and stop shots being taken.
"I knew when I was coming to United this was the style of play. So I was pretty much ready and I was already confident."
