Carlo Ancelotti's side have reached the final of the Copa del Rey and are currently top of La Liga after both Atletico Madrid and Barcelona slipped up last weekend.
They travel to Schalke in the Champions League round of 16 on Wednesday and the 32-year-old believes this could be his side's year.
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"I want to have heard at the end of the season that Real Madrid have won the first treble of their history," he told Marca.
"We know that it's difficult, but we are where we want to be, fighting and competing for all three competitions."
Looking forward to the Veltins Arena clash, Casillas was full of respect for Jens Keller's side, saying his team-mates are anticipating a tough match.
"They're a team with a great physical threat, who are doing well in the Bundesliga and are consolidating their top-four place. They're a dangerous rival.
"We'll have to keep the defensive line strong with the help of other players.
"We have to enjoy the moment. Apart from being a very important game in the second round of the Champions League, you have to enjoy these types of events.
"These are the games everyone wants to play. It seems to me that these games are important and have a global impact."