Carrick has been backed by first-team players Casemiro, Bruno Fernandes, Harry Maguire, Matheus Cunha, Amad Diallo and Kobbie Mainoo in recent weeks as he remains the clear favourite to take the full-time role.
The former midfielder has transformed United since taking over from former manager Ruben Amorim, losing just two of his 14 games as interim manager.
Speaking to Betfair, Shearer stated that if United did not hand Carrick the job after reviving the club then it would be incredibly unfair.
“When Michael Carrick took the Man United role he was being asked to do a job and that job was to improve everything, both on and off the pitch, in terms of atmosphere, but more importantly, getting results and getting them into the Champions League.
“He's done that, and he's done it in a good way, and he's just settled the whole football club down.
“Nothing surprises me in football, but my belief is that it would be really, really harsh, and I know football is harsh sometimes, but I think it would be really, really harsh if he wasn't given the chance to go again, even if it's a one-year contract with different incentives, I just think it would be incredibly harsh for Man United not to give him the job now after the job he's done.”
Last weekend's rousing 3-2 win at home to old rivals Liverpool guaranteed a top-five finish as they return to the Champions League next season where Carrick is highly likely to lead the side back into Europe.