The Old Trafford giants will kick off against Aston Villa on Friday second in the Premier League, just three points behind leaders Liverpool and with a game in hand.
Solskjaer has so far been coy on talk about a push for a first Premier League crown since 2013 but was more forthcoming when it came to the advances made in his two years in the dugout.
The United boss said the players had grown mentally stronger since he had arrived in 2018.
"One, they're two years older than when I came," he said on Thursday. "Two, some of them are really strong personalities that we've brought in and they have really been a good influence.
"I think we have competition for places, which means you cannot go around and think that you can just fake your way through being part of this team because you only deserve to be in a team that you contribute to."
Solskjaer said his players had realised it was a privilege to play for the 20-time English champions.
"When you're here, enjoy it as much as you can and maybe you'll experience some fantastic times," he said.
"We've got winners like Juan Mata and Nemanja Matic, who don't really play every single game but are so influential in the group, in the build-up to games and even during games when they're not playing."
Solskjaer made six changes for the last-gasp 1-0 win against Wolves on Tuesday, when Luke Shaw, Anthony Martial and Scott McTominay came off a bench that included Mata, Fred and Donny van de Beek.
"We have a strong squad," the Norwegian said.
"We've rotated a lot and we have not really found a settled XI this season but I don't think it's about that anyway this season.
"It's going to be about a squad of 25 to 26 players and I think, towards the end of the season, that will help us."
Solskjaer knows about being left out from his time as a player at Old Trafford, when he was often used as a substitute by Alex Ferguson.
The Norwegian scored four goals against Everton in 1999 only to then be benched in their next league game.
"It's horses for courses," Solskjaer said. "Some players you feel are more suitable for some teams but, of course, you do plan ahead.
"I remember those four goals. I remember also the times that you feel you deserve to play and you're left out. That's part of being at Manchester United.
"We have top players everywhere and we know that, throughout the season, if you're going to win something you have to contribute."
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