The Frenchman has won two La Liga titles and two Champions Leagues since taking over the reins in 2016.
But this season has started disastrously as they trail league leaders Barcelona by 19 points.
Real Madrid sit fourth in the table and are at risk of missing out on the Champions League places if they continue their current form.
Stars such as Cristiano Ronaldo and Karim Benzema have not hit the heights of previous seasons and now Zidane has come under increasing pressure to turn results around quickly.
According to Spanish outlet Diario Gol, the 45-year-old has the option to leave the Spanish capital in the summer.
It is reported that Zidane’s national side France could offer him their head coach position after the World Cup in Russia this summer.
Current boss Didier Deschamps’ future will not be decided until after the tournament.
Real will then look at either Germany manager Joachim Low or Chelsea’s Antonio Conte to replace Zidane.
Zidane won the World Cup with his national team in 1998 and would join an exclusive list if he was to replicate that as a manager.
However, in the meantime he must focus on helping Real bounce back from their defeat to Villarreal at the weekend.
Los Blancos are next in action against Leganes in the Copa del Rey quarter-final tomorrow (8.30pm).