Solskjaer was sacked at the Theatre of Dreams in November 2021 following a disastrous run of results that culminated in a 4-1 defeat at Watford.
Carrick was placed in interim charge by United as the club looked to appoint a manager for the remainder of the season.
The former United midfielder took his first steps into coaching under Jose Mourinho following his retirement and he remained at the club when Solskjaer took over.
Carrick spent 15 years at the club in total as a player and coach, and the interim role was his chance to take his first steps into management.
But Solskjaer has revealed that Carrick came to him ten days after his appointment and told him that he didn’t want the role due to his respect for the Norwegian.
‘At United, I’d sometimes see Michael talk to the players,’ Solskjaer told the Athletic.
‘Some of these were ex-team-mates of his which isn’t easy, but he had an authority about him. I am 100 per cent sure that Michael will be the manager of Manchester United if he wants to be.
‘I love him as a person, but while he never said anything to me about it, the day I got booted, I went to see him and said: “I’m gone, they want you to take over.” I told him that I wanted him to do it. After ten days, Michael came to me and said: “I don’t want to do this”.
Carrick was in charge for three matches, winning two and drawing one.
The former England midfielder is now thriving in his first job in management with Middlesbrough, who he’s guided to the Championship play-offs this term.
Solskjaer believes Carrick has all the credentials to become United boss in the future.
‘Michael is a man of value and principles, a big family man, but his knowledge is also second to none,’ continued Solskjaer.
‘He’s had Sir Alex Ferguson and Jose Mourinho as managers, and he’s played with the best players in the world. He’s a winner but he’s still in control of his emotions.
‘I cannot see him not being the Manchester United manager.’
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