The driver was arrested at the scene, the source added.
"At the moment we have no information on the motives of this individual," the source said.
The incident took place at around 9:00 am (0700 GMT), one source said, asking not to be named.
The Renault car first drove at speed towards a bus stop in the city's 13th district, leaving one person seriously hurt, before targeting another stop in the 11th district, causing one fatality.
Julien Ravier, mayor of the 11th and 12th districts, told BFMTV news channel that the victim was a woman in her 40s who was waiting alone at the stop.
The police source said the man was in his mid-30s and was not from Marseille. Another source said the vehicle was a van.
BFMTV reported that a bystander noted the registration number of the car, which the police used to trace the vehicle to the city's Old Port district, where the suspect was arrested.
Police sealed off the port area and urged residents in a tweet to avoid the neighbourhood which is popular with tourists.
The police sources, who asked not to be identified, did not say whether the incident was being treated as a terror attack or an accident.
The incident comes as police across Europe search for the driver of a van that mowed down pedestrians in Barcelona last week, killing 13.