The tabloid claims Liverpool are "bracing themselves" for an approach, with the reigning European champions said to be impressed by Klopp's personality and strength of character, tools required for the job.
Germany national team manager Joachim Low has denied any interest in filling the new vacancy at Real Madrid.
MARCA report that early talks have taken place between the former Argentina defender and Madrid, and while Spurs hastened to make clear that Pochettino has no buyout clause in his newly signed contract, few clubs have deeper pockets than Real Madrid.
However, the Argentine went on to distance himself from the rumours, saying: "I have just signed a long contract with Tottenham and I am very happy here".
"I am not the best coach tactically, but I have other things", he said ahead of the final. "I enjoy what is happening and what has to be, will be", he told Spanish newspaper AS. "[Ex-Newells Old Boys coach] Jorge Griffa told me once that I must let football take me along, not insist on changing my own fate".
Last week, Pochettino said that he was honoured to extend his stay in North London and that he's looking forward to leading Spurs into their new stadium next season.
The former Espanyol player and coach was open to discussing in basic terms the challenge which would face the new Madrid coach, taking over a squad full of very experienced and confident stars who have in the past had issues with past managers including Rafa Benitez and Jose Mourinho. I know that he was offered that position many times by the current president.
"Even before tactical ideas, it is about finding a football identity", he said.
Pochettino, in Spain for the launch of his book "A New World", signed a new five-year deal at Tottenham last month but has been installed as the favourite to replace Zidane at the Bernabeu. Leadership is important in the development of the game.
On Thursday, the Frenchman announced his decision to leave the Spanish giants after just two-and-a-half seasons and just five days after becoming the first manager in history to win three consecutive Champions Leagues.