The former Liverpool defender believes United boss Erik ten Hag is destined for the sack despite getting his side into the FA Cup final.
The Red Devils needed penalties to beat Championship outfit Coventry on Sunday, having let slip a 3-0 lead with 20 minutes to play.
And had it not been for a tight VAR call in extra-time stoppage time, United would have been on the wrong end of the greatest FA Cup comeback.
With United set to miss out on Champions League football, Carragher believes the disappointing display at Wembley will prove to be the last straw for Ten Hag.
He told Sky Sports: “I think that result has just cost Erik ten Hag his job. I think that result has just cost Erik ten Hag his job.
“It’s just poor performance after poor performance. That dressing room is probably the most embarrassed dressing room from a team that has just won a big game.
“You can’t even celebrate that win. Maybe that can somehow win the final but I just don’t see any progress.
“This is somebody’s job we’re talking about, but I just don’t see how they [the owners] can look at the situation and be positive about it. I don’t see it.”
Despite United fans also expressing their outrage over the performance, Ten Hag has instead hailed his side for reaching back-to-back FA Cup finals.
He remarked: “We have reached two finals in two years and that is a huge achievement.
“I see the mistakes we make. You can’t avoid them. But it’s not an embarrassment, it’s a huge achievement.
“We can play at very high levels but in same match we can go to very low levels and it’s not explainable. It’s about game management, responsibility and decision making.
“We have to do better in these occasions. On several occasions like this we let the result slip away but today we got away with it.
“I don’t think heads dropped, but we had many setbacks and maybe our reaction is human but it’s not acceptable. When you are in control you should win easily.
“We can’t say this group doesn’t have the right mentality because they showed character today, but in the final part there was a lack of discipline. This season has had many ups but also many downs.
“Winning the FA Cup doesn’t make the season successful. We want to win more. I have mixed feelings and there are mixed feelings in the dressing room.
“First you have to put yourself in a winning position but then you have to take it over the line.
“In these moments we are making mistakes and not sticking to our responsibilities.”
United will now face rivals Manchester City on May 25 in a repeat of last season’s FA Cup final where they will be out for revenge following their 2-1 defeat.
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