Immigration officials are continuing to question 20 people, including two children, rescued from an inflatable boat off the Kent coast on Sunday.
Those on board, 18 of whom are believed to be Albanian migrants, reportedly alerted their families in Calais after their boat started taking in water.
They were rescued by the UK coastguard, but concerns have been raised that capsize tragedies seen in Greece or Italy could occur in the Channel.
Two Britons are among the 20 held.
A Home Office spokesman confirmed a woman and two children were aboard the inflatable boat that was rescued.
He said a second vessel, believed to be linked to the inflatable that got into trouble, was discovered on the beach at Dymchurch.
‘Not doing enough’
Councillor Mary Lawes, UKIP group leader on Shepway District Council, said she was concerned for the security of the region as well as the safety of migrants seeking to cross the Channel in unsafe boats.
She said: “We are not doing enough to control our coastline, the government has to address border controls, something has to be done to protect these people from harm and our borders.”
David Monk, Conservative leader of the local authority, said he believed high levels of surveillance in the English Channel would mean most boats crossing the Channel would be identified.
He said: “I am pretty sure our security is good.
“I cannot recall a previous incident but this should act as a warning to the authorities to be even more vigilant.”
The UK coastguard said it was called just before midnight on Saturday to an incident off the coastal village of Dymchurch.
A helicopter from nearby Lydd and two lifeboats with coastguard rescue teams from Dungeness and Folkestone began the search.
About two hours later, at 02:00 BST on Sunday, a rigid-hulled inflatable boat, known as a rhib, with 20 people on board was found.
After being rescued, the group were handed over to the UK Border Force and taken to Dover.
The incident comes after 17 men, thought to be Albanian migrants, were detained after a catamaran arrived at Chichester Marina in West Sussex on Tuesday, along with a 55-year-old British man wanted on suspicion of murder in Spain.
The Briton, who was the subject of a European Arrest Warrant, was detained on suspicion of facilitating illegal immigration and the 17 men were held on suspicion of entering the UK illegally.
Also last month, two Iranian men were found floating in a dinghy in the English Channel.