Levy, known for his tough negotiating style, has often been a source of frustration for fans who blame him for failing to pay the wages that would deliver players capable of delivering Spurs' first English top-flight title since 1961.
But the 62-year-old Englishman was also responsible for transforming the North London club with a new stadium and a state-of-the-art training ground.
Under Levy's leadership, Spurs claimed their first major trophy since 2008 by beating Manchester United to win last season's Europa League, earning a spot in the Champions League for the seventh time during his tenure.
Levy became chairman in February 2001 after ENIC acquired a majority stake in the club.
"I am a little bit in shock, because he was my former president. I don't know what happened there," United States manager Pochettino told reporters ahead of Saturday's friendly match against South Korea in New Jersey.
"Of course, I sent a text to him, and I sent a text to different people who were very close to him, and still no answer.
"I don't have more information and only what I want to say (is) I wish him the best in his life, family, time.
"And yes, I was in shock like many (people) that really know him."
Pochettino was in charge of Spurs from 2014 to 2019, transforming them into perennial top-four Premier League contenders while guiding them to the Champions League for the first time, where they finished runners-up to Liverpool.
He was fired five months later, but the 53-year-old Argentine had previously said he remained on good terms with Levy.
"We can't forget our relationship now. Then was a very important period in our careers and for the club also," Pochettino added.