The violence on Monday night occurred in Clichy-sous-Bois and Bobigny, deprived suburbs northeast of Paris, but no injuries were reported, the source said on condition of anonymity.
The confrontations appeared to have been sparked by false reports of individuals driving a white van who had attempted to kidnap young women or children in the area.
“Rumours about kidnapping children with a van are completely unfounded. No abductions have been proven,” the Paris police service wrote on Twitter on Tuesday. “Do not share this false information, do not incite violence.”
It was unclear where the claims originated — one version refers to a white van driving between the suburbs of Nanterre and Colombes west of Paris — but authorities have been battling to stop it spreading in the Paris region.
On March 16, a gang of around 20 young people attacked two people in a white van in Colombes, leaving both of them with light injuries, the police source explained.
Many of the rumours appear to have been spread on the Snapchat messaging service, as well as on Facebook where posts from people claiming their children or family members had been approached by strangers or abducted have been widely shared.
In the Seine-Saint-Denis region, where Monday night's violence took place, the local police chief had issued a warning to officers, urging them to be “careful because there's a psychosis that is starting to set in.”
In December, police in the town of Versailles, 20 kilometres (12 miles) south of Paris, also issued a warning about rumours being spread online about attempted abductions.
In a statement on Twitter, Versailles police reminded users that under French law spreading a false rumour could be punished with fines of 45,000 to 135,000 euros ($50,000 to $150,000).
The FranceInfo public broadcaster also reported in late 2017 how a photo of a white van had been shared thousands of times in the Loiret and Loir-et-Cher regions of central France, again referring to attempted abductions.
“This van is driving around Beaune-la-Rolande and children are being forced to get inside! If you see it, call the police and share this as much as possible!!!” said the accompanying text.
The fact-checking service of the broadcaster said versions of the same false story had been reported from across the country, from the Pas-de-Calais region in the far north to Nimes in the south.
Social media companies are under increasing pressure from governments to help stop the spread of fake news or unfounded rumours which often have real-life consequences.
In India, false reports of kidnappings spread on the WhatsApp messaging application last year led to the deaths of around 25 people who were lynched by angry mobs.