Brazil's Infectious Disease Society director Estevao Urbano told Bloomberg he was concerned about the effect on the healthcare system.
"For the hospital system, just the fact of having an increase in influenza and coronavirus is already a risk," he said. "Most cases will not require hospitalisation, but if the volume of infections is very high, a small percentage is already enough to overload hospitals."
There has been at least six cases of "flurona" reported in the country.
A 16-year-old boy, who was vaccinated against both the flu and Covid, tested positive to both in Rio de Janeiro, according to local media. However, his mother said he was experiencing "virtually no symptoms".
Rio de Janeiro's municipal health secretary Daniel Soranz said 17 potential cases were under investigation but none had become seriously ill, according to EFE news agency.
The Israel case has also reportedly been suffering "mild" symptoms including having trouble breathing.
"The disease is the same disease. They're viral and cause difficulty breathing since both attack the upper respiratory tract," Professor Arnon Vizhnitser told local media.
He said hospitals in the country were seeing more pregnant women with either Covid or flu, pointing out the two viruses can present very similarly.
"This is especially (difficult) when you do not know if it's coronavirus or the flu, so you refer to them the same. Most of the illness is respiratory."
Australia's former deputy chief medical officer Nick Coatsworth said it was not yet known whether the combination of the two viruses was more severe.
"We don't always know whether that makes things more severe or not, but the message, for pregnant women in particularly, you're certainly at increased risk of going to hospital, going to intensive care and having an adverse outcome for your unborn child if you're not vaccinated," he said.
"So the critically important message from 'flurona' is that we've got a vaccine for both Covid-19 and influenza, and we should be taking them."
Demand for the flu vaccine in Brazilian pharmacies had reportedly increased 44 per cent in the past two weeks.