United moved closer to qualification for next season's Champions League with a 2-1 victory over Brentford in the Premier League on Monday (27.04.26) and the former Liverpool defender expects Carrick to be rewarded for the "sensational results" he has overseen since taking interim charge following the sacking of Ruben Amorim in January.
Carragher said on Sky Sports: "He's going to be the Manchester United manager, certainly next season, no doubt about that - and you can't say he doesn't deserve it.
"They are absolutely sensational results. They're the results of a title-winning team, or a team going for the league.
"Now I know that pressure isn't on Manchester United right now, they haven't got European fixtures, all these other things will come into it, but I don't think anybody could have come in and done any better results-wise.
"The biggest criticism of Ruben Amorin was that he never adapted, Carrick has shown he can adapt."
United could seal their return to the Champions League this weekend but Carrick doesn't think that the club should "over celebrate" getting back into Europe's top competition for the first time in three seasons.
The former Red Devils midfielder said: "We've given ourselves a great chance, we've put ourselves in a big position. It's a tough league, it's tough to get wins. You can see that throughout the league. To get the amount of wins we've had. I think the lads have got to take a bit of credit for that.
"The Champions League is one thing, but it's not something that we should be over-celebrating either.
"We want to be finishing high up the league really and we want to be challenging high up in the league and trying to get more points so our season doesn't get to a close when that happens."