Over 18 months since he left Liverpool to take up a role as Red Bull's global football director, the German is being linked with Xabi Alonso's job at Real Madrid.
Defeat at home to Celta Vigo on Sunday has increased the pressure on Xabi, who is struggling to build a relationship with the Real dressing room and also getting his system across.
El Mundo says Real's board are already discussing replacements, with Klopp featuring prominently.
A crisis meeting was held after the Celta defeat, where both Klopp and former Real coach Zinedine Zidane were raised as potential caretaker hires. The summit ran until 1am at the Bernabeu.
It's suggested should Real suffer a heavy defeat to Manchester City in the Champions League on Wednesday night, Xabi will be removed.