"The 12 members of the delegation include players and members of the backroom staff," said a statement from the Health Secretariat in Brasilia, where Sunday's game will take place.
The South American Football Confederation (CONMEBOL) made the diagnosis on Friday night and informed local health authorities.
"All are asymptomatic, isolated in individual rooms, and are being monitored by the CONMEBOL team and local health officials," the statement added.
Venezuela are due to take on reigning champions Brazil on Sunday evening at the Mane Garrincha stadium.
The Copa America, the oldest tournament in international football and featuring 10 South American nations, was moved to Brazil at the last minute after Argentina pulled out because of a COVID-19 surge there.
Brazil agreed to host the tournament in spite of its own coronavirus issues. Nearly 500k people have died in Brazil, more than any country in the world bar the United States, with 2,216 reported dead on Friday.
The Copa America is due to run until July 10.