Klopp is a favourite of Real president Florentino Perez, who is now actively seeking a replacement for Zinedine Zidane.
The Frenchman, while tied to Real until 2020, is expected to be shown the door as his team battles to stay in touch with the top four this season.
Don Balon says Perez has Klopp and Germany coach Joachim Low both high on his shortlist, but the former has no interest in the job.
While he'd love coach Real Madrid, Klopp has told agents he wouldn't consider leaving Liverpool for a job so dominated by football politics. He is also suspicious of Perez interfering in team selection and transfer policy.
Instead, Low and Tottenham boss Mauricio Pochettino are the two early favourites to succeed Zidane in 2018.