The French regional maritime authority said 27 people had died. French officials had previously stated there were 31 deaths but the death toll was revised down, with no immediate explanation for the discrepancy.
A joint search and rescue operation by the French and British authorities launched after a fishing boat spotted people in the sea off France was finally called off late on Wednesday.
French interior minister Gerald Darmanin said the dead included five women and a girl while two survivors had been picked up and were being treated in a French hospital. One of the dead women was later reported to have been pregnant.
Downing Street said they had agreed to "keep all options on the table" in their efforts to break up the human trafficking gangs responsible for putting desperate people at risk in one of the world's busiest sea lanes.
Immigration compliance minister Tom Pursglove confirmed that Mr Johnson had renewed a previous offer to send UK police and Border Force officers to mount joint patrols with the French.
The aim is to prevent migrant boats from attempting the perilous crossing but the French have previously resisted amid concerns about the implications for their national sovereignty.
Mr Pursglove said, however, the last incident showed the two countries needed to deepen their co-operation in dealing with the issue.
"The Prime Minister and President Macron have had exactly that discussion this evening. That is something that I am very keen to see happen," he told BBC2's Newsnight.
"It is the case that in the past we have offered to host and to help with joint patrols. I think that could be invaluable in helping to address this issue. I really do hope that the French will reconsider that offer."
There was shock and dismay on both sides of the Channel at what was widely described as a "tragedy".
The French authorities have arrested four suspected people traffickers in connection with the incident while the regional prosecutor has opened an investigation into aggravated manslaughter.
Following a meeting of the Cobra emergencies committee, Mr Johnson said it was clear that French operations to stop the migrant boats leaving "haven't been enough" despite £54 million of UK support.
He said the people traffickers were "literally getting away with murder" and that he hoped the French would now find the renewed offer of joint patrols "acceptable".
"We've had difficulties persuading some of our partners, particularly the French, to do things in a way that we think the situation deserves," he said.
"I understand the difficulties that all countries face, but what we want now is to do more together - and that's the offer we are making."
However the mayor of Calais, Natacha Bouchart, said that it was the British who were to blame and called on on Mr Johnson to "face up to his responsibilities".
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