Carrick is on the verge of leading Man United to Champions League qualification after taking over from Ruben Amorim earlier this year.
United have won nine, drawn two and lost two of their 13 matches since Carrick's arrival as he revives the club who now look full of life.
Is Carrick right for Man Utd?
United were sixth when Amorim was sacked, and now they are close to securing European football. Despite this, many have their doubt about Carrick including Neville who spoke on The Overlap Fan Debate this week.
“If he beds the foundations in over another year - good principles, good culture, the club happier again, the staff trusting him because everyone would trust Michael Carrick and love him - you could then go and make a bigger appointment. That could be applied.
“There aren't that many things happening up there right now (in the top 6) - it is as poor at the top as it has been for a while. Who is the best person to go and try to exploit that potential window? And is Michael the right person for that, or is it someone else? That's the confusion I've got.
“I was adamant, four months ago before Ole, Ruud, and Michael were all interviewed for the role: please, whatever happens, do not appoint any of them at the end of the season. Bring one in now, stabilise the club, then put someone world-class in who has a great coaching pedigree. That's where I was three or four months ago. Why am I changing now just because of some results?”
Has Carrick had it easy?
Scholes also weighed in on the subject and suggested that Carrick may have had an easy schedule so far at the club.
“I don't know. It's not straightforward, is it? From what's happening now - just playing Premier League games - to the start of next season is a totally different challenge. Suddenly, you're playing three cup competitions, playing Saturday-Wednesday. That's a big deal for him in terms of getting the job next year, and he should maybe question himself seriously about whether he wants it.
“They've got to get players in as well. If they can get (them into) Champions League football, I think that'll enable us to move forward.”
A final decision on Carrick is yet to be made but with just four games remaining, he is by far the favourite to take the role where he has thrived so far.