The incident took place on the famous Promenade des Anglais during a firework display, media reports say.
One image on Twitter showed about a dozen people lying on the street, some being tended to.
The local prefecture has urged people in the area to remain indoors, calling the incident “an attack”.
Nice Mayor Christian Estrosi said that “a lorry driver appears to have killed dozens of people”.
He urged residents to stay indoors.
Some reports spoke of shots being exchanged between police and the occupants of the lorry but these have not been confirmed.
Social media video showed people running through the streets in panic following the incident.
A journalist with the Nice Matin newspaper reported from the scene that there was “a lot of blood and without doubt many injured”.
An Agence France-Presse reporter said the incident took place as the firework display was ending, adding: “We saw people hit and bits of debris flying around.”