The incident took place at two different areas of the city at around 09.15 a.m. local time (0715 GMT).
Police arrested the driver, according to local newspaper La Provence. However, it is still not clear if it is an accident or a deliberate attack.
Local police tweeted "an operation was ongoing" and asked people to avoid the city’s Old Port area.
BFMTV broadcaster reported the detained man was a 35-year-old suffering psychological problems. He is reportedly known to police and was driving a stolen car.