Spain international Casillas has been with the European champions his entire club career, but the goalkeeper's future appeared in doubt in Mourinho's final season in charge at the Santiago Bernabeu in 2012-13 when he was dropped to the bench in favour of Diego Lopez.
Initially, Ancelotti kept faith in Lopez and deployed him as his number one in La Liga matches last term, but he departed to join Milan in the close-season.
Casillas has subsequently enjoyed a renaissance and has firmly re-established himself in the Real side.
The Real captain admits he endured a "love-hate relationship" with Mourinho, but, despite confessing his preference for Ancelotti's tenure, he is adamant that any future meeting with the now Chelsea manager would be amicable.
"Ancelotti is better than Mourinho," he told Cadena Cope. "With Mourinho I had a love-hate relationship.
"In his time here I did not speak of him when he was in my locker room, nor do I think that I have to talk now when he is already managing another team and I'm with my team. It would be a bit self-seeking and little cowardly on my part.
"[I hope] that things go very well for him. In the time that he was at Real Madrid, he won titles and you have to give him thanks.
"If I cross paths with him tomorrow I am going to extend my hand, then it will be up to him if he shakes it. I think he would shake."
Casillas, though, stated that his relationship with former team-mate Xabi Alonso has deteriorated
The duo are thought to have fallen out over Alonso's public backing for Mourinho in the past, while the now Bayern Munich midfielder endorsed Manuel Neuer to win the FIFA Ballon d'Or ahead of Real superstar Cristiano Ronaldo - causing some in Spain to accuse Alonso of a lack of respect.
"It does not interest me what he says. In the last years our relationship has not been adequate," Casillas added. "Obviously Cristiano Ronaldo deserves the Ballon d'Or.
"Everyone has their opinion. If Xabi has that opinion, mine is different."
