The 22-year-old was criticised by his Portuguese manager last April for a lack of commitment to the club.
"He's a few months free of these minor injuries that sometimes stop evolution and I'm really happy, so I think the natural consequence of it is that he will have his contract and he will be a Manchester United player for years."
The United manager played down previous criticism of the player.
"With Luke it is not about changing my mind. With Luke, it's just the evolution or no evolution of his potential because I know Shaw since he arrived in the Premier League with Southampton and I know his potential and his quality.
"Could he come in my direction, in the way I think football, the way I like my players to be not just on the pitch but also during the training week? And he made a big effort."