The spokesperson said there were several “lifeless bodies” after the tragedy, which occurred during a crowd panic, but was unable to confirm how many boats and people may still be in danger at sea.
The boat had 110 people on board, Reuters said, quoting Northern France authorities.
Around a hundred migrants have already been rescued by a French navy tug, and are being taken to Boulogne, French newspaper La Voix Du Nord said.
Emergency services picked up the father of a four-year-old girl, who was “in tears on the beach”, the paper said.
It comes hours after the UK’s government’s controversial Rwanda bill – intended to deter migrants from crossing the Channel in small boats – was passed.
Home Secretary James Cleverly, whose department is responsible for the effort, said on X on Tuesday: “These tragedies have to stop. I will not accept a status quo which costs so many lives.”
He said ministers are “doing everything we can to end this trade”.
On Tuesday morning, Sky News filmed a suspected migrant boat attempting to cross the Channel. We do not know if it is the boat which has failed to make the crossing. The latest updates on the migrant crisis is here.
Sky’s Europe correspondent, Adam Parsons, in Dunkirk, said at least four vessels were working off the coast of Wimereux as well as helicopters and there were “reports of survivors”.
Sea conditions could not be blamed, he said, as they were “perfect”.
He said: “If you were trying to cross the Channel in a small boat, this is the day you would do it, so if you can’t make it on a day like this, it shows how dangerous it is.”
The Channel is one of the world’s busiest shipping lanes and currents are strong, making the crossing on small boats extremely hazardous.
People smugglers typically overload rickety dinghies, leaving them barely afloat and at risk of being lashed by the waves as they try to reach Britain.
The Rwanda bill, which Rishi Sunak says will curb the illegal trade, will finally become law after the House of Lords decided they would no longer oppose it following hours of wrangling. The measure was finally approved just after midnight.
For weeks, peers have been pushing back on the scheme – which seeks to deport asylum seekers arriving in the UK via small boats to the African nation – and trying to get ministers to make changes to the controversial legislation.