His public criticism of Mourinho's tactics led to the decision with Ashley Young leading United out in recent weeks in Antonio Valencia's absence.
Mourinho is under pressure at United to turn around the club's fortunes after a dismal start to the Premier League season.
According to Calciomercato, the United board are already preparing for life after Mourinho, and have contacted Allegri's entourage to discuss whether the Juve manager would be interested in moving to England.
Carrick expects a proven performer to deliver further success at United if given the necessary time and support, with there few coaches in the modern era who can claim to have matched his record. "This is my opinion and I would have said that to him, I would say "no, that's not something to say because you are not the manager".
Arguably, Pogba has begun the season in his best-ever form for the Red Devils at a time when the team is failing to click as a whole and lie in eight place in the Premier League table as a result.
"He's managed to win trophies at United", said Carrick.
Pogba believes being in the World Cup-winning France team is in itself a big achievement and he never plays to get the armband.
"I think he is the right coach for that club, for that team".
The two of them really don't appear to get along particularly well, with passive aggressive comments in the press and a video of a training ground argument fuelling speculation that one will move on soon enough.
"I don't think there's a right or wrong, I think it depends what fits at the right time for that individual and that's where I'm at right now".
"I'm in my position as I am at the moment and it feels good for me, so we'll see what the future brings".