Shaw has put his Old Trafford troubles behind him with a sparkling start to the season earning himself an England recall.
The defender missed the narrow win at Watford on Saturday due to being concussed against Spain at Wembley earlier this month.
However, Shaw is expected to start the Champions League tie at Young Boys on Wednesday.
The 23-year-old left-back has another year to run on his deal and his return to form strengthens his hand during negotiations with the United top brass.
Mourinho told Sky Sports: “I hope they will stay.
“Luke Shaw's evolution makes us believe that he is going in the right direction, I want to keep him.
“We had a similar conversation last season about Marouane Fellaini and I was always saying, I hope he stays, I believe he will stay, and I say the same thing, I hope they stay.
Mourinho was also full of praise for Young, Mata and Herrera wanting all three, alongside Shaw to extend their stay at the club beyond next summer.
He added: “Juan Mata and Ander Herrera, even if they are not playing a lot or if they are not starting matches, by the character point of view and by what they represent in the modern society in football they are almost unique pieces that I also don't want to lose.”
United have rallied with back-to-back wins at Burnley and Watford but after their trip to Switzerland face Wolves at home on Saturday in the Premier League.
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