Carrick has taken over from interim manager Darren Fletcher who stepped in for Ruben Amorim who was sacked 2 weeks ago following the 1-1 draw with Leeds United and his comments against the club hierarchy.
The former midfielder is set to leave at end of the season, with the club planning to bring in a permanent head coach which includes names such as Oliver Glasner and Andoni Iraola.
Carragher was asked about Carrick and did not hold back however, suggesting that he was not elite enough to lead the Red Devils in the long term.
"I am not going to change my mind," Carragher asserted, referring to the post-derby hype. "I always think this when a Liverpool manager is appointed: Do I believe this guy can win us the league? It doesn't mean he will do it, but I have to believe he can.
“I don't believe Manchester United can win the league with Michael Carrick as manager, or Chelsea can win the league with Liam Rosenior as manager.
You have to be really special. Think of the managers who have won the Premier League, 99 per cent of them are special, and I think Michael Carrick lacks that. I might be proven wrong, but that is why I will take Michael Carrick out of that."
Carragher has put together a four-person shortlist that Manchester United should be looking to appoint their next manager from which included Julian Nagelsmann, Thomas Tuchel, Eddie Howe and Luis Enrique who remains at Paris Saint-Germain.