Mourinho has been out of work since being sacked by Manchester United towards the back in December.
However, the Portuguese football manager has remained in his west London home and is out enjoying the Wimbledon build-up at Queen’s.
Murray is competing in the doubles draw at Queen’s alongside Feliciano Lopez, just five months after undergoing hip surgery.
And the former world No 1 has received the backing of Mourinho, who watched him practice and sat courtside during his quarter-final match.
“In this moment of his career where he did so many things, it depends on your DNA and your mentality,” Mourinho told Amazon Prime.
“Sometimes in football, some guys in the last years of their careers just give up. They just think what I did is enough, my history is big enough, I don’t need to sacrifice myself to play more years.
“I think he is all about the passion and love for the game and to try to overcome all the difficult moments.
“I am far from being an expert but I was watching him in his practice session and for me it was a positive surprise.
“He was moving well and it was a positive surprise for me. If he manages to enjoy for a few more years, apart from the titles he can win or not win, if he can enjoy and not suffer in his future then I think it is fantastic.”
Mourinho and Murray struck up a close friendship when the former was manager of Chelsea.
He allowed the tennis star to use his club’s facilities during his recovery from injury.
“He lived very very close to the Chelsea training ground,” Mourinho added.
“When I was at Chelsea Andy had some problems and he needed some medical support or something to use the facilities and we were always open to that.
“This time I was not supposed to come but I always felt I was behind him in the big moments, the big tournaments, the big finals.
“I think this is probably a more important moment than all these ones.
“What he is doing is fantastic so I think it is nice for him the people who love him and support him are always for him.”
