Mourinho accused certain pundits and reporters of being 'compulsive liars' in response to criticism of his handling of Rashford, pointing out that the 20-year-old featured 52 times for the Red Devils in all competitions last season.
Liverpool legend Carragher claims that Mourinho is missing the point, however, and fears that Rashford is going down the same route as fellow United academy product Danny Welbeck, who was shipped out to Arsenal four years ago.
"I agree with Jose Mourinho in terms of his appearance stats. They are very good," he told Sky Sports News. "What I don't like is that Jose or other people felt I was just throwing away a lazy comment. The reason I said what I said is because I am talking about Rashford in terms of a centre forward. For all the stats Jose gave, they are spot on.
"But the one he didn't say was that since Jose has arrived, Rashford has started at centre forward only 12 times in the Premier League. Rashford has said he wants to be a striker. Now he may have changed - he may be happy playing on the left or the right, or staying at Manchester United. But my whole point was that it's the most difficult position to come through at a top-six club.
"I'm not saying that Jose is wrong and should play Rashford over [Romelu] Lukaku. He shouldn't. Lukaku is better than him. But if that lad wants to play centre forward, how does he do that? If Rashford doesn't want to play centre forward then stay, but if he does then I feel he may have to move away. I like Danny Welbeck as a player, and I hope this doesn't come across disrespectfully - Marcus Rashford does not want to become the next Danny Welbeck.
"He came through at United as a centre forward, then he was used on the left and right and was delighted to get minutes. But then he ends up as a squad player and can never nail down a position. He moves to Arsenal and the exact same thing happens - where does he play? He is still a squad player and does the same for England. That's not what you want."
Rashford is currently serving a three-match ban after being shown a straight red card in United's 2-0 win at Burnley.