Around 900 more infections were recorded in the past 24 hours, taking the total to 5,400 since January, Veran told France 2 television.
Separately, the national health agency Sante Publique France, which has been updating its tally daily, put the new toll at 127 deaths.
The sharp increase came as France held the first round of nationwide municipal elections Sunday, expected to show a record abstention rate as virus fears kept many people from polling stations.
Veran said government officials would meet with scientific advisers in the coming days to determine if the second round of voting will be held next Sunday.
"Probably on Tuesday I will ask them about the second round," Veran said, while refusing to be drawn on rumours the government will soon announce a complete lockdown and confine people to their homes.
"You hear all kinds of things, but if someone is going to sign such an order, I'll know about it," he said.