United are in a strong position to secure second spot in the Premier League table this term, while they will take on Chelsea in the FA Cup final on May 19.
The Red Devils were disappointedly knocked out by Sevilla in the Champions League round of 16, however, and have been unable to challenge Manchester City for the league title.
De Gea has admitted that "it has not been quite enough this season", but the Spanish goalkeeper is predicting a more successful 2018-19 campaign.
"It has not been quite enough this season. Fair enough we are battling to hang on to second place and hopefully we will fire until the final game to ensure that happens, but we have not managed to win our initial target which was to win the Premier League title," De Gea told Sky Sports News.
"That is what we are all here for, that is what the players at the club want and also to give a good account of ourselves in the Champions League but we have also not managed to do that.
"Obviously the FA Cup is very important, a nice trophy to win, and we are going all out to win that. But it is not quite enough - we are Manchester United and have to aim for bigger and better things. That has always got to be our focus."
United are in Premier League action away to Brighton & Hove Albion on Friday night.