Mourinho won three trophies during his three-year stint at Santiago Bernabeu, although the latter part of his tenure is best remembered for a public fall-out with Casillas which led to the club captain being frozen out of the first team.
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"I've only been training with Carlo, I don't know what differences there are [between Ancelotti and Mourinho], but it's not a question of looking back. You have to be grateful for Mourinho's contribution to Real Madrid, but now we have to start with the hope that the club can be at the top," he told a press conference.
"He's a winner, that's obvious to everyone, but he'll make it clear from the beginning. This club was on the edge of titles last season; we want to complete our objectives in all three competitions."
Casillas says competition for places at the club is good under Ancelotti - but admitted he was desperate to regain his No.1 spot next season.
"Competition is always healthy. There are very good goalkeepers here and we must learn to live with that. I hear a lot about competition for my post, but in other parts of the pitch there are two or three players for one position as well," he added.
"I would love to be No.1 keeper. I work for it. But if you ask the others, they're going to say the same. You have to compete to play in the XI of Ancelotti. I will have to fight as I always have."