United have 1 win, 2 draws and 2 defeats from their first five games in what is their joint-lowest points tally after five games in the Premier League era.
The Red Devils are already exploring managerial alternatives according to reports, placing immense pressure on Carrick who could be sacked if results do not turn around.
United sit 12th in the table, have been knocked out of the EFL Cup by Brighton and looked clueless tactically despite their strong end to last season under Carrick.
Speaking on The Overlap Fan Debate, Carragher backed Carrick and stated that it would be ridiculous to say goodbye to a manager who led the side back to the Champions League.
"I think it would be ludicrous to get rid of Michael Carrick.
"I don't see Michael Carrick as the Manchester United manager in the long term. I almost feel like he's in there doing a job for the club to get them through a rocky period and then passing the baton on.
"If that sounds disrespectful to him, I don't mean it like that. He came in, and Manchester United needed calm, they needed not being a circus, not being a banter club.
"His results have been unbelievable since he's come into the job over the long period. He got the job on the back of that. The fact he got them in the Champions League brings the money.
"Manchester United need to be a Champions League club again. It's not about, wow, we're going to win the league this year. They need another season of just making sure you get the Champions League, and then it'd be a case of bringing in one of the top managers, about who could manage a club at that level.
"I don't think the idea of changing Michael Carrick now or at the next international break makes sense. It just needs to be relaxed, just let him get on with it."
With United having scored eight goals and conceded eight so far in the Premier League, they must produce a result against Tottenham after the international break. A win is imperative for Carrick or he may face the chopping block from an impatient board.