United go up against Tottenham on Sunday in what promises to be a blockbuster Premier League showdown.
Solskjaer’s side were beaten 6-1 by Mourinho’s team earlier in the campaign, in a result that sparked a big reaction in the aftermath.
But the Norwegian has steadied the ship at Old Trafford in the time since, guiding his United side to second in the table.
Mourinho previously claimed that, when he steered United to the silver medal in 2018, it was one of his biggest achievements as manager.
Yet Solskjaer has now taken issue with those words, saying: “If we manage that you can’t say that is an achievement. Our ambition is a lot more than second place.
“I said that when I wasn’t even at the club – you can never be happy with that and we’re not.
THINK YOU KNOW SPORT? Test your sporting knowledge with our tricky quiz
“The mood in the club is good, but the players are not happy – they are hungry to improve and win something.
“They want to establish themselves in and around the top of the league.”
The United boss also wants his side to make sure they finish second playing good, exciting and convincing football along the way.
“If you look back at the last six or seven years, we have not been stable in the top two or three, the top positions,” he said.
“There have been ones when we’ve been fifth, sixth or seventh. So if we can get a good second place, not just scrape into it at the end of the season but have a good finish.
“Then we will go into the next one with hopes we can go even further and do even better.
Alex Telles ‘to be used in Man Utd swap deal’
Chelsea may have signed perfect Haaland partner
Lionel Messi’s Barcelona anger explained
“We know we’re in a good position at the moment but don’t get me wrong, our ambition is to catch the team in front of us.
“We’re in the Europa League quarter-finals and have had a good first leg win against Granada in Spain.
“We’ve had some more good results in the league as well, we’re in second place and fighting to get that, and hopefully also shorten the distance to Manchester City.
“Right now you cannot give a rating on our season until it has finished because it can be defined in three days, it can be so quick.
“It can be the difference between going into a final, having a strong finish in the league or just doing okay and scraping over the line, which is such a different scenario.
“So I will tell you at the end of the season – hopefully on May 26. Then we can see the progress we’ve made.”
United’s 6-1 defeat to Tottenham was a huge surprise earlier in the campaign – but Solskjaer insists it was understandable given their lack of fitness at the time.
“We had such a low level of match fitness at that time with so little preparation for the season,” he said.
“So even though we scored with an early penalty we were just not at the level where we could compete especially when Anthony was conned into that red card.
“That was also a big turning point as well, but we learnt a lot from it, especially not to get wound up, and not to react.
“So even though the players will all be fired up for Sunday, I still want to see a lot of composure.”
Solskjaer recently said ‘trophies were for ego’s’.
And Mourinho, true to form, has taken aim at the Norwegian over those comments.
“It’s his opinion. I’m pretty sure his big boss Sir Alex had a different opinion about it. But I respect what Solskjaer thinks. I think different,” said the Tottenham boss.
“Freedom of thinking, freedom of ideas, and if that is what he thinks that is fine by me. It’s not a problem for me, he’s not being disrespectful to anyone.
“But I repeat I believe his big boss, and the biggest in Premier League history, has a different view in relation to that.”