The French interior ministry said in a statement that French patrol vessels found corpses and people unconscious in the water after a fisherman sounded the alarm about the accident.
Police then said in a statement that “more than 20” people had died.
Three helicopters and three boats have been deployed to take part in the search, local authorities said.
Interior Minister Gérald Darmanin, who is heading to the scene, wrote on Twitter that “many people” had died in the incident, adding that “the criminal nature of the smugglers who organise these crossings cannot be condemned enough.”
“The disaster in the Channel is a tragedy,” added Prime Minister Jean Castex. “My thoughts are with the many missing and injured, victims of criminal smugglers who exploit their distress and misery,” he wrote on Twitter.
The disaster, the worst single loss of life recorded in recent times from migrant crossings in the Channel, comes as tensions grow between London and Paris over the record numbers of people crossing.
Britain has urged tougher action from France to stop migrants making the voyage.
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According to the French authorities, 31,500 people attempted to leave for Britain since the start of the year and 7,800 people have been rescued at sea, figures which doubled since August.
Seven people have been confirmed dead or are still missing feared drowned after various incidents this year.
French police said this week they detained 15 suspected members of an international migrant smuggling syndicate that helped people illegally cross the Channel to Britain.
The network of Iraqi Kurds, Romanians, Pakistanis and Vietnamese helped a minimum of 250 people per month cross to England, using small boats that transport up to 60 migrants at a time.
Passage to England would cost a migrant €6,000 and the smugglers racked up some €3 million in total profits.
According to British authorities, more than 25,000 people have now arrived illegally so far this year, already triple the figure recorded in 2020.