Madrid go into Saturday's clash at home to Atletico third in La Liga, eight points behind their derby rivals and 11 adrift of leaders Barcelona.
The champions lost at Real Betis last weekend as their hopes of defending their crown receded further and the team was jeered during Tuesday's Copa Del Rey win over Alcoyano at the Bernabeu.
But Casillas told the club's official website: "We have absolute confidence in Mourinho.
"He is a coach that has given us a lot. He took a very young team up against a team like Barcelona, which was winning lots of titles, and has managed to bring them to a similar level.
"Every time there is a match between the two teams now there is no favourite like Barcelona were before.
"The coach 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.
"He has had some very good seasons. Being the best coach in the world is very demanding and perhaps that's the problem that he has.
"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.
"Memories are short in football and maybe people do not remember who took the team to a record-breaking league title, the most goals, the most points. We must also remember the good times.
"This team does not want to stop and wants to achieve more."
This season, however, it is another capital club who have emerged as Barca's closest challengers with Atletico having won 11 of their opening 13 league games.
And Casillas knows the hosts have to close the cap on the top two if their are to remain contenders.
"It is an exciting match that people want to see, especially the opponents' fans because they are coming into it playing well, but for us it is a vital match," the Spain keeper said.
"We have to do what is needed and that is to win and to keep believing. We have a match against an opponent who is above us and if we beat them we the cut points difference."
Should Madrid get a result, Barcelona have the chance to extend their lead at the top when they host Athletic Bilbao.
The visitors won their rescheduled Europa League clash in Israel against Ironi Kiryat Shmona yesterday as unbeaten Barca cruised past Alaves in the cup.
David Villa took his career goals total past the 300 mark in that game, playing the full 90 minutes as he continued his recovery from the broken ankle he suffered last season.
"I hadn't played a whole match for a long time and I'm very happy to have lasted the full 90 minutes," the striker said on the club website.
"I've made the recovery and I feel great. Since Monday I've been training without any ankle problems."
Elsewhere on Saturday, fourth-placed Malaga, who are just four points behind Madrid, travel to Getafe and Valencia welcome Real Sociedad.
On Sunday, Javier Aguirre takes charge of Espanyol for the first time as the league's bottom club travel to relegation rivals Granada.
Former Mexico boss Aguirre has replaced Mauricio Pochettino, who was sacked on Monday, and will try and earn the Catalan club only their third league win of the campaign.
They go into the game two points behind their opponents and victory could lift them out of the bottom three.
Another struggling side, Deportivo La Coruna welcome Real Betis, while sixth-placed Levante head to Celta Vigo and Real Mallorca host Real Zaragoza.
This weekend's round of fixtures starts on Friday night with second bottom Osasuna at home to Rayo Vallecano, while on Monday, Sevilla host Real Valladolid.
Friday's La Liga Fixtures
Saturday's La Liga Fixtures
Sunday's La Liga Fixtures
Granada vs Espanyol
Deportivo La Coruna vs Real Betis
Celta Vigo vs Levante
Mallorca vs Real Zaragoza
Monday's La Liga Fixture
Sevilla vs Valladolid