Skipper also hails David de Gea
Reports this week have suggested that the former Chelsea boss has grown frustrated at the Spanish giants and will leave at the end of the season.
The goalkeeper told Onda Cero radio: "Jose Mourinho wants to stay and Real Madrid wants him to stay. I can say this as I'm the captain.
"We have a young team and he knows in the long run this will pay off. He loves the city and he loves the club. We know that Madrid has to achieve more right now. But remember, we're a young and new-look team."
Spain's World Cup winner also offered some thoughts on Atletico Madrid stopper David de Gea, the 19-year-old who may yet inherit Casillas' mantle as Spain's No.1.
"What De Gea is doing is bloody incredible, and we have to be grateful to coaches who give these youngsters a chance," he concluded.