The Red Devils are chasing a Champions League spot in the Premier League thanks to Carrick who has led the side to an unbeaten start this season with wins over the likes of Arsenal and Manchester City.
United beat Everton 1-0 on Monday night away from home, pushing them back into the top four as the side chase down European football.
Despite such an excellent start under Carrick, Neville admitted that he may not be the right choice when the season concludes.
"United have to pursue another manager right now around the fact that you don't know how the results are going to go.
"If he gets United into the Champions League there will be a lot of fever around that should be appointed. I'm not against him being appointed, I love him to bits. But I think United should go for the best class of manager available.
“They've had young and inexperienced managers in the last two picks. Ole Gunnar Solskjaer has come in before as an ex-player and that didn't work in the end. So I feel removing as much risk as possible is the right option."
Former Real Madrid boss Zinedine Zidane has been linked with the United role alongside the likes of Oliver Glasner and Andoni Iraola as the board weigh up their options ahead of the summer.