They have also won the FA Cup, Europa League and League Cup once during his stay.
“I don’t think he will be happy until he wins more of the bigger trophies. Staying at Manchester United in this time and not having won more, I think that’s something that he will feel is a black dot in his career,” Solskjaer told British media.
“I don’t think David is that type who cares about personal awards, he wants the team to lift the big ones as well.”
De Gea is set to become just the second goalkeeper after Alex Stepney to make 400 appearances for United.
“He has come out and said that he wants to play many, many more – maybe 10 more years,” the United manager said. “How long he will be able to stay at the top is just up to him.”
United are fifth on 58 points, a point and a place behind Leicester City. They host Southampton later on Monday.