Shaw is about to embark on a make or break year at United.
"Because he is a fantastic guy in my two years and with Mr. (Louis) Van Gaal before me", Mourinho said, as quoted by FourFourTwo.
The Armenia global said: "United was hard but you have to be ready for everything so I kept going".
But speaking on United's tour of the US, he revealed he has made a decision to stay unless Jose Mourinho tells him he is not wanted.
There have also been question marks over the 23-year-old's conditioning, as he enters the final 12 months of his contract at Old Trafford.
Shaw has endured injury, notably a broken leg in September 2015 that ended his season.
It was initially thought that Martial, who has expressed a desire to leave United this summer, received authorisation to dip out of the club's USA tour base in Los Angeles to be with his family.
Darmian has attracted interest from a slew of Italian clubs with Juventus believed to be in pole position, but Inter are also striving to take the defender back to his homeland.
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Asked if he doubted his future after he was substituted at half-time of United's FA Cup clash against Brighton and Hove Albion last season, Shaw replied: "Not doubts, but there has been emotion".
"But I've worked hard on my fitness over the summer, not just because of the criticism and the pictures, I wanted to get in the best shape and that wasn't my best shape". The manager was probably thinking: 'Jesus Christ!'I said to my girlfriend: 'I've just opened it and I must look so weird now.' It was fine.
Despite Mourinho's public criticism of Shaw, the England worldwide said the Portuguese coach texted him in the off-season to spur him on.
"I will be [staying] unless something changes dramatically and the manager comes to me and says I'm not in his plans". Now I know what I need to do.
"It was a breath of fresh air when he texted me", he added. Shaw, though, says he's just got a big body. He says these things because he knows I can do it.
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That doesn't mean to say that he hasn't still got a lot to offer at the very highest level, though. I'm used to it. "It still gives me confidence".