The 22-year-old Spaniard has overcome an inconsistent start in England to establish himself as the Red Devils' No.1 shot-stopper this season.
De Gea's recent impressive form had even led to suggestions that Barcelona could target him as Victor Valdes' replacement, but the former Atletico Madrid star says he is happy in Manchester.
He told Mundo Deportivo: "I have a contract [at Manchester United] and they are my team. You must do well in every game, not just against Real Madrid, and that is my intention.
"It is always nice to be linked with Barcelona but I am happy in Manchester with one of the best clubs in the world. Each day I am more happy.
"At the moment [I am not thinking of leaving]. I am in a good stage of my career and I hope to continue like this. Barcelona know that they have Victor who is a great goalkeeper and they want to keep him!"
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"It is not just my name that appears. At Barca, Valdes is still the keeper and Madrid is my city, one day I'd like to return.
"But now I hope to stay in Manchester for a long time and win a lot of titles."
The Madrid-born shot-stopper went on to reveal that he is feeling "settled" in Manchester after a turbulent start to his life in the English top-flight.
"It is going very well. I am very happy. Now I am more settled, more mature and have a lot more confidence," he continued.
"[It] is normal. If a young goalkeeper comes from outside England it is always difficult. You come to a new league with new people and the language... Everything is difficult.
"It is difficult to adapt but, thanks to God, I have adapted, everything is going well and I hope to be there [Old Trafford] for many years.
"It is good for keepers [to go to England]. You learn a lot and improve in all aspects. The Premier League is tougher, more competitive and in that regard it is good to come here for keepers."
The current Spain Under-21 captain also highlighted the influence of Sir Alex Ferguson on his improved form, describing the Scot as a "father".
"Sir Alex is more than a manager. He controls everything in the team. He's a father for all those who come to the club. For that he is the best," De Gea added.
"A footballer has to get used to good and bad words.
"At times things are said that don't make a lot of sense and it is part of the job to know how to live with that. The best thing is to know who you are always."
Meanwhile, De Gea says that without Nani's red card in their Champions League tie against Real Madrid, Manchester United would be in the quarter-finals of the cup.
He continued: "Sir Alex has a lot of character and on that day he was very angry, although deep down what he felt more was pride for his team.
"We were superior to Madrid and deserved more but there were situations - unfair decisions - and we were knocked out.
"Football is like that and in that type of game details leave you knocked out. The expulsion of Nani marked the game and the tie. Without that red card it would be Manchester United in the quarter-finals."