Carragher was one of several former England stars claiming the 20-year-old should leave United in order to develop as a centre-forward last week - citing his impressive form on the international stage.
However, Mourinho responded with a detailed list of statistics backing his development at the Red Devils in correspondence to his playing time.
And while Carragher noted Mourinho's defence, he insinuated the Portuguese tactician missed the point of the message, while stopping short of blaming the manager or Rashford himself of the current dilemma.
"All of the stats Jose has listed are spot on," Carragher wrote in his Sky Sports column. "But another stat that is spot on is that Rashford has only started 12 times as a centre-forward in the Premier League.
"He burst onto the scene as a striker and said he wants to be a striker. I like Danny Welbeck as a player and I don't want to come across disrespectful, but Marcus Rashford does not want to become Danny Welbeck
"What I mean by that is this: he came through United as a forward, and then he was used on the left and the right and was delighted to get the minutes. Then he ends up becoming a squad player, never nailing down a centre-forward position.
"He moves to Arsenal, the same thing has happened, he's still a squad player. And it's the same for England. That's what you don't want," the Premier League football expert added.
"That's the problem you've got as a young player. If Rashford doesn't want to play centre-forward, he should stay. If he does, I feel he'd have to move away.
"Rashford is so high profile because he burst on the scene, we thought we had another Michael Owen, Robbie Fowler or Wayne Rooney in the England squad. That's why there's more talk of him because he's more high profile and he plays more."