An individual had made an official request to police to hold the demonstration about the film that has sparked protests across the world but was refused permission, the source said.
If the individual tries to hold the march he faces six months in jail and a fine of €700 ($900), the source noted.
France's interior minister has said he will ban all protests over the low-budget film "Innocence of Muslims" after a violent demonstration last weekend near the US embassy in Paris.
But social networks were awash this week with appeals for Muslims in France, home to western Europe's largest Islamic community, to defy the ban and hold fresh protests in Paris, Marseille and other major cities.
Tensions were heightened when the satirical magazine Charlie Hebdo on Wednesday published obscene cartoons that mocked both the film and the Prophet Mohamed.
The protests that have left over 30 people dead in the last week have, until now, largely been targeted at the United States, which has had to carry the can for the California-produced "Innocence of Muslims".
But French ministers fear the focus could shift to Paris's overseas outposts following publication of the cartoons featuring obscene images of the founder of Islam.
Embassies, consulates, cultural centres and international French schools in around 20 Muslim countries will be closed on foreign ministry orders Friday for fear of retaliatory violence following weekly prayers.