Rashford has struggled to consolidate a solid starting position since Jose Mourinho took over Manchester United in 2016 and is in the pecking order behind Romelu Lukaku.
This is partly because Mourinho plays him in a variety of positions, although the 20-year-old would rather have a central attack.
Although he scored twice for England in the international break, Rashford was the subject of relentless speculation about his future.
Some bookies have suggested that Chelsea and Everton are the most likely targets should Rashford leave Old Trafford.
As he enters the final year of his deal next summer, the attacker will have to make a difficult decision.
And now it seems that Rashford is ready to change things at United.
A United source said The Sun: "Marcus loves the club and does not want to move, but he wants to be a striker or at least play the middle as No 10.
"He will not give up proving himself to United, but he knows he'll make a big decision soon."
When asked if Rashford should leave Old Trafford, Carragher said, "I think so, yes.
"To play in the middle for Manchester United means you have to be world-class – Rashford is not yet world-class – he could be 23 or 24."
"So maybe Rashford has to move away to return to a top team.
"The problem he has, and so is every young striker in big clubs, nothing is more difficult than this position.
"And you look at Lukaku, I do not see him ever driving Lukaku out while he's there."
Rashford will play the first game of a three-game ban when United play Watford on Saturday (5:30 pm).