United won 2-0 against Tottenham on Saturday to extend Carrick's 100% start to four games in what is an incredible run after taking over from former manager Ruben Amorim.
Reports suggest United have already begun the process of hiring a permanent boss, and Carrick will be near the top of that list if he can secure Champions League football for next season.
Carrick has transformed United into a side who can climb back into Europe, and Hargreaves spoke to TNT about how the club cannot ignore his obvious success so far.
"I don't see how you couldn't give it to him. I know they had the experience with Ole, but he finished second and did great right up to a point. In the end in football you get the highs and lows. When it's good, you’ve got to enjoy it.
"Right now, Man United have it good. So if he finishes third or fourth, from where they were and all the noise, how could you not?
"The only thing you could throw at it was experience, and if you could get a Luis Enrique or Tuchel, maybe. But right now, the most important thing is not the manager or the players, it's a football club winning, and the football club is winning, and Carrick is the architect of that. So why would you change that?"
Even though the Red Devils have been eliminated from both domestic cup competitions, but Champions League qualification would represent a successful season that Carrick could use to leverage a full-time job with the side.