The England squad for the 2016 European Championship in the summer will be announced on Monday.
Rashford, who is just 18 years of age, has been in good form for United in recent matches and has been showcasing his quality.
West Ham striker Carroll, who has earned nine caps for the Three Lions’ so far in his career, has also been among the goals in recent weeks.
Former Newcastle striker Shearer has suggested that if he had to pick between Rashford and Carroll, he would go for the former.
“I think Rashford’s more of an option than Andy Carroll,” Shearer told The Sun. “He’s got great potential.”
“His energy, pace and when he’s been given the chance in front of goal, he hasn’t let anybody down,” added the Newcastle legend. “He is a more realistic option to replace Welbeck.
“I also think referees don’t look favourably upon the likes of Andy Carroll at international level. Especially in major tournaments, there’s foul after foul after foul…”
Rashford has scored four goals and created six chances in 10 Premier League appearances for United this season.
Carroll has scored nine goals and created 20 chances in 26 Premier League matches for West Ham this campaign.
United are fifth in the league table at the moment, while the Hammers are currently sixth in the standings.