The 20-year-old scored the winner for England against Switzerland on Tuesday night, but has so far not started a game for his club this season.
Carragher believes Rashford may need to leave United in order to get regular football if he is to become a world-class player.
Carragher told Sky Sports: “Romelu Lukaku is in front of Rashford and I don’t see Rashford displacing him as long as he is there.
“But remember when Lukaku was at Chelsea and he had to come away and go to Everton. Then he ended up as top scorer and got his move to Man Utd.
“Possibly Everton is the sort of club you are looking at, just below the top six, for someone like Rashford, similar to what Lukaku did.
“You know you are going to play every week as the centre-forward and you know if you have a couple of games where you don’t score then you are still going to play the next week.
“Last season after they played Brighton away, Mourinho had a little pop at him and Lukaku went straight back into the team. That’s the problem you will always have.
“Does he have to leave United? I think so. Is he good enough to displace Lukaku? Is Rashford good enough to be Manchester United’s centre-forward for them trying to win the Premier League or the Champions League? He may not be.
“It gets to the stage where you want to play every week. Will he be an England regular or be Manchester United’s real centre-forward where you know, when the team is picked every week, that he is in it. Will it ever get to that stage?
“In that position you have got to be world class. Rashford is not world class yet, but he might be at 23 or 24, as Lukaku has done. He looks like one of the best strikers in the world.
“Rashford may have to move away to come back to one of the top teams.”