The Red Devil got the only goal of the game during England’s 1-0 victory over Switzerland in Leicester on Tuesday, steering home a Kyle Walker cross at the back post, but has found his chances limited at Old Trafford.
Carragher does not see that situation changing rapidly and has advised the 20-year-old to take a step back to go forwards.
“Rashford may have to move away to come back to one of the top teams,” he told Sky Sports.
“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 Romelu Lukaku has done. He looks like one of the best strikers in the world.
“Lukaku is in front of Rashford and I don't see Rashford displacing him as long as he is there.
Carragher outlined his thinking, stating that consistent opportunities in the first team are crucial.
“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,” he said.
"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.”
Rashford has yet to find the mark at club level this season, having turned out three times for United in the league.