The head of United's academy, who won 39 caps for the national team, feels selecting the 18-year-old for the tournament in France could be a move that backfires.
Roy Hodgson has not ruled out the prospect of selecting Rashford, who scored four goals in his first two club games, but has not netted in four appearances since then.
Asked if Rashford would be ready to play at Euro 2016, Butt told BBC Sport: "I think that's well too soon for Marcus.
"It could backfire on everybody and might not be the right thing for him.
READ MORE | 'Mourinho rumours unfair on Van Gaal'
"I think talk of England is well too soon. He has phenomenal potential for England and Manchester United and I think [picking him] too soon might do him more harm than good."
Butt insisted he was reluctant to advise national coach Hodgson, but feels Rashford is still a long way from being the finished article.
"I'm not telling an England manager what to do; it's his job to pick whoever he feels," said Butt.
"But for me he's a young boy who's still learning the game and has got a long way to go yet."