Zidane revealed his decision in a hastily-arranged press conference in the Spanish capital on Thursday.
Pochettino has gradually improved Tottenham since arriving from Southampton in 2014, guiding them to three consecutive top three Premier League finishes, and a group stage victory over Madrid on the way to the 2017-18 Champions League round of 16.
The Frenchman took over a divided dressing room after the sacking of Rafael Benitez in January, 2016 and immediately set about uniting the squad.
Firstly, Pochettino's success has been built around a work ethic and mentality not appearing to fit into the ethos of Real Madrid.
The FC panel answer your tweets on all things Real Madrid following Zinedine Zidane's departure and share their picks for the World Cup semifinals.
He ended their five-year wait for a La Liga title last term and though Real fared disappointingly in domestic competitions this year, they became the first club since Bayern Munich in 1976 to win three European Cups in a row.
In the interim, Ronaldo will switch his focus elsewhere, as he gears up to lead Portugal's charge at this summer's World Cup in Russian Federation.
The newspaper reports Real Madrid has made Liverpool manager Jurgen Klopp its number one target to replace Zidane but the Reds are expected to resist any move for the German. The league was disappointing but we won the Champions League. "It ended well for me when I was a player and it's ended well for me as a manager".
Zidane's decision has left everyone in shock.
"You have to accept his decision and just thank Zidane for everything he has done", he said.
After impressing hugely during his first managerial post with Bordeaux he was then appointed as coach of the France national team and guided Les Blues to a quarter-final place in Euro 2012, where they were eliminated by eventual Champions Spain.