He was being treated for the virus throughout the week and was in intensive care.
"Today, all of Madrid is in mourning following the passing of the president who dedicated a huge part of his life to his great passion: Real Madrid," the club said in a statement. "Given the current circumstances, Real Madrid will give him the recognition he deserves as soon as possible."
Sanz's son also confirmed the news on Twitter Saturday.
"My father has just died," he said, via a translation by Alex Kirkland of ESPN. "He didn't deserve this ending and in this way. One of the best, bravest, hardest-working people I've ever known leaves us. His family and Real Madrid were his passion. My mother and my siblings enjoyed all of his moments with pride. RIP."
Sanz served as president of Real Madrid from 1995 to 2000, helping the club win two European titles.
He is one of the latest victims of a pandemic that has killed over 11,000 people worldwide through Saturday, according to the World Health Organization. Spain has had 19,980 confirmed cases, third-most in the world behind only China and Italy.