De Gea surpassed Peter Schmeichel to become the overseas player with the most United appearances after playing his 399th match in Friday's (AEST) 3-0 Premier League win at relegation-threatened Aston Villa.
Despite the landmark moment, De Gea's future has been debated due to a number of mistakes and the form of Sheffield United loanee Dean Henderson this season.
But United manager Solskjaer was full of praise for De Gea, who has won the Premier League, Europa League, FA Cup and EFL Cup among other honours since arriving from Atletico Madrid in 2011.
"It did not cross my mind for one second," said Solskjaer when asked if he had considered dropping De Gea ahead of Monday's showdown with Southampton. "Not at all. To be a Man Utd goalkeeper you have to have a strong mentality. David has a strong mentality and he has been here for so long that he is coming up to 400 games, and you don't do that by being afraid of the limelight or being afraid of Old Trafford.
"You do that by walking out on to that pitch and showing them your quality and mentality and he is a strong character and I didn't see any good in taking him out at all, no."
Solskjaer, whose United has not won the Premier League since Alex Ferguson's final season in 2012-2013, said: "David has been fantastic and since I have been here I have enjoyed working with him.
"He has come out and said that he wants to play many, many more, maybe 10 more years, how long will he be able to stay at the top is just up to him because it's a pleasure to watch him being humble enough to work hard."
"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," continued Solskjaer. "I don't think he will be happy until he wins the bigger trophies, and staying at Man United in this time and they are not having one more, I think that's something that he will feel is a black dot in his career.
"Yeah, he's won his personal awards, 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."
United are in the midst of a 17-match unbeaten streak across all competitions as they look to capitalise on a favourable weekend of results in the Premier League in the race for Champions League qualification.
Chelsea and Leicester City both lost, meaning in-form United can move third in the table with three games remaining if they beat Southampton at Old Trafford.