Wenger has been linked with the job numerous times in the past, given his success at Arsenal in the early 2000s, coupled with the rapid chopping and changing of managers at Real Madrid.
"I am honoured to have signed a new long-term contract as we approach one of the most significant periods in the Club's history and be the manager that will lead this team into our new world-class stadium", said Pochettino.
Real's LaLiga struggles, finishing third and 17 points behind fierce rivals Barcelona, no doubt factored in World Cup victor Zidane's decision.
Zindine Zidane's unexpected departure from Real Madrid has prompted Chelsea to explore the Frenchman as a possible manager.
Zidane stepped down from the role on Thursday, less than a week after Madrid beat Liverpool in Kiev to win a third consecutive Champions League title.
"I've signed a contract and I'm very happy at Tottenham because they let me work and we're growing together". It ended well for me when I was a player and it's ended well for me as a manager.
"I live in the present, there is nothing more important than that", Pochettino said from a book launch in Barcelona. After doing three years, it's the right decision.
"There is no way I think Jurgen Klopp is going to be tempted to move given the season he's just come off".
Zidane said he felt like it was the right time for him and the club to make a change. I'm not looking for another team.
Pochettino, 46, only signed a new contract with Spurs to 2023 last week, so Real would have to pay a big premium if they want to appoint him.
Zidane has already had his dream job in coaching, so it is unlikely he will jump back into management until he is happy with his destination. "I am so happy in Tottenham and I'm so proud that Tottenham showed the commitment and trust in the way that Daniel Levy (Spurs chairman) showed me, and of course, we have a lot of dreams, we have a lot of objectives".