Rashford, 20, has managed 32 goals in 126 appearances for United since making his debut during the 2015-16 campaign.
The attacker also netted his fourth England goal in Saturday's 2-1 defeat to Spain but missed another couple of good opportunities in the English capital.
Shearer has said that he is "a big fan" of Rashford but has admitted that the youngster - mainly used out wide by the Red Devils - will need to leave Old Trafford in order to receive a regular run at centre-forward.
"I am a big fan of Marcus Rashford. He gets into good positions, runs in behind defenders and is a constant threat. His goal at Wembley on Saturday night showcased what he is all about," Shearer told The Sun.
"Unfortunately his two missed chances that could have earned England a point in their Nations League match with Spain also told you where he is at right now. A centre-forward who is playing regularly in that position for his club probably would not have fluffed those.
"Time is not a great commodity in football nowadays. I am not advocating Rashford moves away from Old Trafford. It is a great club to be at. But if he wants to achieve an ambition to be recognised as a great goalscorer it is just not going to happen there.
"He would need to move with the assurance that his next club will make him the main man, the striker, the one up top or one of two. Then I truly believe he would develop into a goalscorer."
Rashford is yet to score in three Premier League appearances for United this season.