More than 230 people have attempted the crossing in small boats since November.
Labour leader Jeremy Corbyn said in a tweet: "We have a duty to reach out the hand of humanity, support and friendship to people who are in danger and seeking a place of safety".
Home Secretary Sajid Javid abandoned a luxury safari holiday to return to the United Kingdom to tackle the "major incident" involving a rise in the number of people attempting the perilous crossing, the Sunday Times says.
Ahead of Mr Javid's cross-government meeting, the NCA warned that more "highly dangerous" attempts by migrants to reach the United Kingdom are likely - and said tackling organised criminal gangs is an "operational priority".
Dover MP Charlie Elphicke has repeated calls for Border Force cutters to be called back from search-and-rescue operations in the Mediterranean to patrol the Channel.
Shadow home secretary Diane Abbott branded Mr Javid's efforts to bring the crisis under control as "flawed" and criticised his slow response.
British interior minister Sajid Javid says the fast-growing number of would-be immigrants crossing the English Channel from France in dinghies is a "major incident".
The group were handed over to immigration authorities after being spotted in Kingsdown, near Deal.
"This can not be done on the cheap with the half-hearted Dad's Army-type set-up suggested by the Home Office in the past", he said.
Shutdown solution seems unlikely before Dems take House
The shutdown impacts 25 percent of the federal government, including the Interior, Justice and Homeland Security departments.
Immigration Minister Caroline Nokes will visit Dover on Saturday to discuss the situation on the ground with border officials.
"We are in close and continual contact with the French authorities and law enforcement partners including through the new U.K. -France Coordination and Information Centre which opened in Calais at the end of last month to tackle criminality at the border", Nokes added.
Some MPs - including Conservative backbenchers - have criticised a lack of detail, as well as calling for more Border Force patrol boats to be deployed.
Mr Javid declared a "major incident" on Friday after dozens of migrants in small boats arrived on the Kent coast over the Christmas period.
He said it was vital the United Kingdom struck "a balance between protecting them (migrants) and protecting our borders".
He warned that more boats might act as a magnet for migrants hoping to be picked up and taken to the UK.
Mr Javid said yesterday: 'The situation in the Channel is of grave concern, with people gambling their lives in reckless attempts to reach the United Kingdom in unsafe boats and treacherous conditions'.
The charity Detention Action said the reason for the crisis was the government's failure to create safe and legal routes for people to claim asylum in Britain.