The former midfielder took over the managerial reins on a permanent basis after a successful second spell as interim head coach last term.
United sit 22nd in the Champions League table after a 4-1 thrashing by Como on Wednesday, while they lie in 13th in the Premier League after they were held to a 1-1 draw by Leeds over the weekend.
They were outplayed by Cesc Fabregas’ men in Italy, as Nico Paz shone at the Stadio Giuseppe Sinigaglia.
Dejected: Bruno Fernandes
Getty
Fans turned on Carrick on social media after the full-time whistle, with the United boss forced to front up the travelling support and apologise himself, holding two arms aloft.
Yet while the majority are calling for his head, Cole, the former Chelsea and England midfielder, believes that the under-fire manager is not at fault for United’s predicament.
“It was humbling for them,” Cole said on TNT Sports.
“The spotlight will fall on Carrick's feet, and he is not the problem, believe me. The club's rot has been there for many many years, and they [the club] need to get out of their own way.
There are a lot of players in that United squad who probably are not good enough to play for United.
“So when does the criticism come to the director of football? I think it's fair that you are trying to get rid of players on long-contracts, but if nobody wants them it's difficult to get rid of them at the right price.”
Fellow TNT Sports pundit Dimitar Berbatov, the former United striker, called the players’ efforts into question.
"What can you say after a performance like this?” the Bulgarian asked rhetorically.
“Today, I don't think everyone was giving their best. It didn't look like this on the pitch. Como were just having fun.”