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England boss Roy Hodgson said Rashford was in his thoughts for France after the striker, 18, scored four goals in his first two appearances for United.
But Butt said: "I think talk of England is well too soon for Marcus."
He said Rashford had "phenomenal potential" but an international call-up now could do "more damage than good".
The 41-year-old Butt, who made 387 appearances for United, said: "It could backfire on everybody and might not be the right thing for him.
"I'm not telling an England manager what to do; it's his job to pick whoever he feels.
"But for me he's a young boy who's still learning the game and has got a long way to go yet."
Rashford scored twice on his debut against FC Midtjylland and added another two goals in a 3-2 win over Arsenal but has since not scored for four games.
Former England international Butt was speaking at a Manchester United event with Gulf Oil International.
Marcus Rashford is still learning the game and might suffer if called up by England for Euro 2016, says Manchester United's academy chief Nicky Butt.