The 46-year-old, who also played for Barcelona, Fernerbahce and Besiktas was rushed to hospital on Saturday after his Covid-19 symptoms rapidly developed.
Rustu was in goal when Turkey finished third at the 2002 Japan/Korea FIFA World Cup and was named in the team of the tournament.
He holds the record for the most appearances for the Turkish national team with 120 caps.
Rustu’s wife, Isil, with whom he has two children, confirmed the sad news on Instagram.
"I would have liked to give you better news while being transparent about the truth but I'm sorry to say that my husband is hospitalized with the diagnosis of Covid-19," Isil wrote.
"Everything was normal when he suddenly developed symptoms really fast and we are still in shock. These are critical and difficult times.
"My only request is for people to have conscious and be respectful. My test was negative along with my son and my daughter. He's in hospital and we are not allowed to see him.
"This is the most difficult part, not being able to be with him. God is great and he's entrusted to Turkish doctors. These days will pass too. Please do not spare your prayers. #YouAreStrongRR."
Turkey has 7,402 coronavirus cases so far with 108 deaths.
The Turkish top flight was one of the last leagues in Europe to be suspended with former Chelsea midfielder John Obi Mikel cancelling his contract with Trabzonspor when it took them too long to do so.
