The Spain goalkeeper is set to make his 400th appearance for the club in Monday night’s game against Southampton.
During that time he has established himself as one of the best keepers in the world – but only has won one Premier League trophy since moving to Old Trafford in 2011.
Solskjaer insists he will live with regret if he doesn’t add to that.
“He’s won the Premier League, he’s won the FA Cup, the Europa League, the League Cup, but that is probably not enough for him,” said the United manager.
“I don’t think he will be happy until he wins the bigger trophies, and staying at Man United in this time and then not having won more, I think that’s something that he will feel is a black dot in his career.
“Yeah, he’s won his personal awards, but I don’t think David is that type who cares about personal awards. I think he wants the team to lift big trophies as well.”