The former England international, who made 464 appearances for United, immediately moved into a coaching role at United after retiring from playing in 2018.
An astute and skilful midfielder, the six-time Premier League winner has had the chance to use his extensive knowledge to improve Solskjaer's men.
Weir had an opportunity to watch and learn from Carrick when he was coming through the ranks at United, making his debut for the first-team in 2016 when the veteran was captain.
And the 24-year-old says Carrick was the best player he trained with and believes his calming presence will benefit the dressing room.
"He is quite a quiet guy, I always remember him having a sarcastic sense of humour," Weir told Tribalfootball.com. "People always ask me who was one of the best players to train with and I always say Michael Carrick, who I reckon is one the most underrated players I've ever played with.
"He's a top player. He's a calming influence, he's very calm.
"So when things get a bit in the media, I imagine he would calm down the lads and have a quiet word with one or two of them on the side. He's a great guy to keep on there and he has great experience as a player."
United players have been vocal about Carrick's presence in the dressing room.
£47m-signing Fred looked on his way out of Old Trafford but has turned things around this season.
And the midfielder credited Carrick for helping him improve his game.
"He knows about the difficulties, the shortcuts, and he tries to show me everything there is in training and in games," the Brazilian said in April, via The Sun.
"He's a guy who has been of huge importance for me at United and I'm very grateful to him."
To read our exclusive interview with Weir about his journey from United to Slovakia, click here.