The teenager has earned eight senior England caps, most recently featuring for Gareth Southgate's team in March, but the FA is reportedly keen for Rashford to collect tournament experience in Poland, something Mourinho does not agree with.
Mourinho told reporters after the Anderlecht victory: "When a player reaches a certain level, it doesn't make a lot of sense to drop levels and play in the under-23s. Let's say in a derby against Manchester City, [United coach] Nicky Butt doesn't pick him to play that game.
"So when you reach a certain level - and he has reached that level - then I don't think it makes sense. But the power is in their hands."
That was a longer night than we had hoped for....— Michael Carrick (@carras16) April 21, 2017
One step closer
Thanks to The Kid pic.twitter.com/kRPa0AbOyp
Injuries to Zlatan Ibrahimovic and Marcos Rojo marred United's progress to the last four of the Europa League and Mourinho accepted he faces tough team selection issues for Sunday's Premier League trip to Burnley.
"[Ander] Herrera plays, he didn't play today," Mourinho said. "I was thinking about him and [Marouane Fellaini] as changes but because of the injury to Marcos Rojo I couldn't do it.
"Herrera is fresh to play and Ashley Young is fresh to play. Who else? Nobody else is fresh to play."
United's record goalscorer Wayne Rooney made his return against Anderlecht, taking his place on the bench after missing four games with an ankle injury, but Mourinho suggested the club captain is not yet ready to play.
"He's not in the best of his condition," Mourinho told reporters. "He's not at the level of sharpness you need to play a high level of football.
"But he has the experience and he has the character."