The Spain international has kept an impressive 21 clean sheets across all competitions for United this season and won the club's Player of the Year award for a record fourth time on Tuesday.
De Gea's efforts have helped second-placed United reach the FA Cup final but the 27-year-old believes the trophy would not make up for a poor showing in the Champions League and failure to win the league title.
"It has not been quite enough this season," De Gea told Sky Sports. "Fair enough we are battling to hang on to second place but we have not managed to win our initial target which was to win the Premier League title and 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 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."
De Gea has called on his team to step up.
"Next season is a new season and we have to improve in a number of areas," the Spaniard added.
"It is not just Manchester City we will be challenging next season, five or six teams fight for the Premier League. It is a case of hard work to be in there at the wire."
United, who need four points from three matches to guarantee finishing second in the league table, travel to Brighton and Hove Albion on Friday ahead of the FA Cup final against Chelsea on May 19.