Some passengers are still missing, and about 50 migrants were rescued, some of them by the British vessels, the authorities said, after earlier reporting one person had died.
Two British ships and several French vessels have been involved in a search and rescue operation, an earlier update from France’s Maritime Prefecture of the Channel and the North Sea said.
The RNLI said a lifeboat crew was launched just before 4am. Search efforts are still continuing.
In Dover, people were seen being stretchered off a lifeboat and surrounded by paramedics as urgent medical treatment was given – although most of those pulled from the sea were picked up by French vessels and returned to France.
Four French vessels were involved in the rescue, named Pluvier, Notre dame du Risban, Abeille Normandie and Cormoran.
A French helicopter was also deployed to the scene, as well as a ship from the RNLI and another British private vessel.
Early on Saturday morning, information was received from a patrol boat that a migrant boat was sinking off Sangatte, French authorities said.
Many of those rescued were returned to Calais aboard the patrol vessel Cormoran, MailOnline reports. Some were also picked up by the RNLI’s Dover ship.
It is unclear how many people were on the boat and whether more are missing at this stage.
