A spokeswoman from French coastal authority Premar said between five and 10 others are missing, AFP reports.
British and French coastguards have rescued around 50 people from the vessel, authorities said.
A number of people were seen being brought off a lifeboat on stretchers in Dover but the extent of injuries and exact numbers remains unclear.
The RNLI said a lifeboat crew was launched just before 04:00 BST on Saturday.
French authorities say they were called to reports of a boat at a "standstill" off Sangatte, near Calais.
The first victim was an Afghan man aged between 25 and 30 years old, a local prosecutor told AFP news agency.
A volunteer, who was on one of the rescue boats, told Reuters news agency migrants were using shoes to bale water out of the sinking boat.
Anne Thorel said there had been "too many" people on the boat.
The English Channel is one of the most dangerous and busiest shipping lanes in the world, with 600 tankers and 200 ferries passing through it every day.
The UK Home Office has been asked for more details about the incident while Home Secretary Suella Braverman has said it was a "tragic loss of life".