United's hopes of qualifying for the Champions League are slim, but De Gea believes that it would be foolish not to compete for a spot in the Europa League.
"There are still plenty of important games to play," the Spaniard told reporters.
"We are Manchester United and it's our duty to win every game we take part in.
"That's what we'll try to do now because we want to finish as high as we can in the league, ensure we're playing in a European competition next season and show we are still around and that we will never give in."
United are currently seventh in the Premier League.