A star of the 2002 World Cup, Rustu played just seven times for Barca - all within the 2003-04 season - but retired aged 39 in 2012 with 120 international caps to his name.
The 46-year-old's wife Isil posted on her Instagram page on Saturday to inform fans of her husband's diagnosis.
"I would like to give you more positive news while sharing the truth in the most transparent way, but I'm sorry to say that my husband Rustu is hospitalized with the diagnosis of COVID-19," she wrote.
"We are shocked by the rapid symptoms that developed suddenly when everything was normal. These are critical and difficult times.
"May Allah give emergency relief to all patients, including my dear husband.
"My only request is that people can be a little more conscientious and respectful in the process. In the tests we have done, my daughter and son were negative and only my husband was positive.
"For this reason, we are at home while he's in hospital, and we are not allowed to see him. In fact, this is the most difficult part, not being able to be with him.
"Allah is great and is entrusted to him and the Turkish doctors. I hope this will pass."