Real hosts Barcelona on Saturday at the Bernabeu.
Real Madrid has won 15 straight in all competitions, including 10 in La Liga, but Pep Guardiola's Barcelona side remains the one club that Mourinho has not solved yet while in charge in Spain.
Barca, the European and three-time defending Spanish champions, is unbeaten in six straight La Liga matches against Real. The run includes a 5-0 victory when the clubs met for the first time with Mourinho in charge of Real.
Guardiola has led Barcelona to seven wins and three ties in 11 matches against Real in all competitions, with the only loss coming in the Spanish Cup final.
This time, Madrid wants better success against Barca at the Bernabeu. And this time, it enters the match with a three-point cushion atop the standings, and a game in hand on Barcelona.
In addition, Karanka believes Real is better suited to defeat Barcelona in its second year under Mourinho.
"We've been together longer now, we know our players better," Karanka said, "we're more confident as a result of this and are earning good results. This helps the team be in good form."
Barca playmaker Xavi wants the game to be "open, with both teams going for it, where one can see the potential of both teams." But exactly how each side will approach the match remains an unknown.
Mourinho solved previous Barcelona teams with his counter-attacking style, but that was with Inter Milan and Chelsea. With just a single win against Barca at Real, despite having Cristiano Ronaldo on board, it may be time for a change.
"We're only concerned about focusing on our own team, using our own system and showcasing our principles, which will remain the same," Karanka said.
What is know is that Barca will have the best player on the field in two-time world player of the year Lionel Messi. The Argentine has 13 goals in 15 games against Real, while Ronaldo has just three in nine meetings.
Messi, Xavi and Ronaldo are the three finalists for this season's Ballon d'Or, awarded to the best player in the world, but the lone focus this week will be on a good, clean match.
Real has been shown numerous red cards in its recent meetings with Barca, and it will set a new club record winning streak if Mourinho picks up his first win in the series in La Liga.
Even if that happens, who knows if Mourinho will talk about it.
Also Saturday, Levante hosts Sevilla and Real Betis hosts Valencia in the pair of undercards.
On Sunday, Rayo Vallecano hosts Sporting Gijon, Getafe hosts Granada, Athletic Bilbao hosts Racing, Villarreal hosts Sociedad, Malaga hosts Osasuna, Espanyol hosts Atletico Madrid, and Real Zaragoza hosts Mallorca.