Witnesses described hearing gunfire, although this was not immediately confirmed by the authorities with images on social media showed casualties lying in the street, some being tended by paramedics.
The incident comes as France remains badly shaken by November 2015 attacks which killed 130 people at the Bataclan concert hall in Paris and nightspots across the city.
Describing it as a "major criminal attack", Sebastien Humbert, the sub-prefect of the Alpes-Maritimes region, told BFMTV the driver had left "several dozen, perhaps around 30 people dead" and "around a hundred injured".
An AFP reporter described seeing a white truck driving at high speed onto the famed Promenade des Anglais as people were leaving after the annual Bastille Day celebration display.
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"We saw people hit and bits of debris flying around," he said.
Terrified pedestrians screamed as they fled the area. "It was absolute chaos," he added.
The incident came just hours after France announced that a state of emergency declared after last November's attacks would come to an end later this month.
Nice city mayor Christian Estrosi urged residents to stay in doors.
"The driver of a van appears to have killed dozens. Stay in your homes for now. More info to follow," Estrosi wrote on Twitter.
Local authorities from the Alpes-Maritimes prefecture also urged people to remain indoors following what appeared to be a deliberate attack.