The retired Old Trafford boss spent 26 years in charge, winning 38 trophies.
He said: "He must be the proudest man in the world. He replaced the biggest coach in English football. Now it is up to him. He has to coach, he has to decide who to buy, who to play, he has to train them every day.
"He has worked so hard for so many years at Everton, waiting for a big chance. Now he has a big chance.
"I cannot wish him very, very well because he is competing against me.
"But if I was in Spain at this moment I would wish him very well because he deserves good things."
Mourinho then joined the debate over whether Moyes' lack of silverware during his career could be a disadvantage.
"When he finished fourth with Everton or fifth, it's a trophy. He doesn't take the medal home but it's a trophy.
"So I think he has trophies and I hope he gets some with Man United. But not many because we want some for us."