The Rio de Janeiro-based club said in a statement that 293 employees were tested, including all players, their families and people working at home. The club said 38 tests came back positive, but did not identify any of those infected.
A staffer who spent 40 years with Flamengo died on Monday of COVID-19.
Brazilian President Jair Bolsonaro has insisted soccer resume during the pandemic. More than 8,500 people have died because of the coronavirus in Brazil.