The split over Mourinho, under pressure after his side slipped 11 points behind leaders and arch rivals Barcelona in La Liga, was evident as some supporters whistled when others chanted his name in a 3-0 second-leg win against third-tier Alcoyano that sent Real through to the last 16 7-1 on aggregate.
"I think a couple of months ago the people that were whistling yesterday were the people who were clapping and cheering when we won the league," Casillas said.
"Memories are short in football and maybe people do not remember who led the team to win the record breaking league title: most goals, most points," added the Spain international.
"We must also remember the good times and the continuity in this project, which started a long time ago, with a very young team that has been improving.
"We've won the Copa del Rey, La Liga, the (Spanish) Super Cup. This team does not want to stop and wants to achieve more."
Mourinho has a chance to win over more of the club's demanding fans when city neighbours Atletico Madrid visit the Bernabeu in La Liga on Saturday.
Atletico, resurgent under Argentine coach Diego Simeone, are second, three points behind Barca and eight ahead of third-placed Real.
"We have absolute confidence in Mourinho," Casillas said.
"He took a very young team against a team like Barcelona which was winning lots of titles and has managed to match the levels of the two teams," he added.
"Yes it may be being unfair to the coach. He arrived at a new team which was very young three years ago and I think much of Real Madrid's success has been because of him.
"Being the best coach in the world is very demanding and perhaps that's the problem that he has."