The incident took place on Tuesday night off the coast of Pas-de-Calais, the local prefecture posted on X.
Two children are believed to be among those who died, according to reports.
It comes after a woman was confirmed dead on Tuesday afternoon by Kent Police after attempting the crossing.
Border Force vessels and RNLI lifeboats had been responding to small boats in the Channel on Tuesday.
The translated post from the French authority on Wednesday morning said: "A new tragedy occurred last night off the coast of Pas-de-Calais. Three migrants lost their lives, three other people are likely missing.
"The prefect was at the port of Boulogne-sur-Mer this morning, alongside the mayor of Le Portel and the prosecutor of Boulogne-sur-Mer, to oversee the security and rescue operations.
"Smuggler networks bear the responsibility for these tragedies. The State is determined to combat them."
The prefecture said the three people who died were trying to reach the UK, while the other three migrants went missing during a separate attempt to make the dangerous journey.