Real paid AC Milan around 70 million euros for Kaka, now 29, in 2009, a year before Portuguese Mourinho joined from Inter, but he has failed to reproduce the form that earned him the World Player of the Year award in 2007.
Recent media reports suggested he might be sold or loaned out but he told the club's TV channel on Thursday the goal he scored off the bench in Sunday's 6-0 demolition of Real Zaragoza appeared to have quashed speculation about his future.
"It gives me hope because there was a lot of talk regarding me leaving Real Madrid this summer and it seemed to put an end to all that," he said.
"Mourinho believes I can improve and help the team. It's good to know he's aware that I'm by his side. I'm going to fight to continue to be there, but also to be a starter."
Mourinho said last week he was happy to have Kaka in the squad and the Brazilian said he was grateful for the vote of confidence from the former Chelsea manager.
"He's always had faith in me," Kaka said.
"He could have said he didn't want me anymore, that the club should look for another team for me.
"He believes I can help the team considerably. The rest of the coaching staff does too.
"I hope to enjoy football and to find personal success at Real Madrid. The fans and I want to see the best Kaka in action. That's what I wish for this season."
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