Two adults and a five-year-old child are reported to have died, according to French media, but authorities have declined to comment.
At least 18 people have been taken to hospital.
The UK's home secretary Priti Patel has been briefed on the situation.
Two people have already died making the perilous journey across the English Channel this year, including one earlier this month.
Military resources and civilian boats have been involved in the rescue operation after the vessel was seen in difficulty near Dunkirk.
The people on board are thought to have been trying to cross to the UK, despite wind gusts of up to 28km/h.
The alarm was raised by a yacht which notified search and rescue.
French patrol boats and a helicopter from the Belgian air force were dispatched as well as a fishing boat.
Eighteen people have already been taken to hospitals in Calais and Dunkirk, according to the Maritime Prefecture of the Channel and the North Sea.
The search operation is continuing.
An investigation into the causes of the sinking has been launched by the Dunkirk public prosecutor.
Read our full response to today's tragic news that at least three people, including a young child, are feared dead after a boat carrying 20 refugees capsized in the English Channel.https://t.co/lW0FuXOMoA
— Care4Calais (@Care4Calais) October 27, 2020
Care4Calais founder Clare Moseley said the refugee community in Calais is “utterly devastated”.
She said: “We are grieving for the victims, we stand in sympathy and solidarity with their families and friends.
“It is cruel and horrifying that this time, young children are among the victims.
“This unnecessary loss of life has to stop. No one should ever feel they have to get into a fragile craft and risk their lives crossing the Channel, least of all vulnerable children.”
She called for the incident to be a “wake-up call” for those in power in the UK and France.
Dover MP Natalie Elphicke, who has been outspoken on the issue of Channel crossings, tweeted: “It is terrible that tragedy has struck in the Channel again. People traffickers have no regard for life, no matter how old or young.
“These perilous crossings must be stopped, once and for all, before there is more loss of life in these stormy winter seas.”