Mourinho stated in October that it "made no sense" for Rashford to be called up for the finals in Poland given that he is an established part of the senior England set-up.
There are also suggestions that the United boss is concerned at the prospect of the 19-year-old missing a whole pre-season for his club, with Rashford having struggled for consistent form following his explosive debut campaign of 2015-16.
Rashford says the final decision over his potential inclusion in Aidy Boothroyd's squad will be out of his hands, but he has made it clear that he would enjoy the challenge of more tournament football.
"It is up to the coaching staff and the managers to decide if I go to that," he told BBC Radio Five Live. "But if I do go, I'll be approaching it with the same attitude as I do to the seniors. It is exciting times.
"Getting tournament experience is massive, as long as everything is aiming towards the long-term objective, we obviously want to be successful and win trophies. So whatever we have to do to get to that, whether it is going down to gain the experience, everyone is happy to do that."
Rashford has only scored three Premier League goals this season - the last of which came on September 24 - and has been more often deployed in a wide role to accommodate Zlatan Ibrahimovic as United's central striker.
But the forward insists he has developed into a stronger all-round player under Mourinho this term.
"I'm a completely different player now compared to where I was this time last season," he said. "People don't always look at that, they look at the results but they don't look at what you do to get the results. When it all comes together people will start to realise.
"My overall game [has changed]. I've been playing in wider positions and now me knowing what the forwards wants helps me, and also helps me know that the winger wants when I get in the middle.
"The modern-day forward and attacker is not just a number nine or number 10 or just a winger. It is someone who can interchange between the positions if they are needed to. I think I'm a number nine but if I have to move to other positions for certain games it is what you have to do and it is part of your attributes."