The senator said via Twitter that he was fine, had no symptoms, and was being treated at home with hydroxychloroquine and azithromycin.
Flavio is the fourth member of the president's family to be infected by the coronavirus.
President Bolsonaro was diagnosed with COVID-19 on July 7, but tested negative just over two weeks later after working from home.
On July 30, the country's First Lady Michelle Bolsonaro tested positive. She announced last week tests showed she was virus-free.
Ten days ago, the president's youngest son, Jair Renan Bolsonaro, 22, also tested positive for COVID-19.