UK police and social services have called for urgent help to deal with the impact of the Calais migrant crisis.
The leader of Kent County Council has met Home Office officials to request support in dealing with the arrival of hundreds of young migrants in Dover.
The backlog has grown as migrants make fresh attempts to enter the tunnel.
More than 3,500 attempts have been made this week to get into the Channel Tunnel, with people gathering at fencing at its freight terminal.
'No more capacity'
In the UK, Highways England said there were nearly 6,000 lorries parked on the motorway as part of Operation Stack, which will continue into the weekend.
It is the first time Kent Police have asked neighbouring forces in south-east England to help deal with the chaos.
County council leader Paul Carter said a "massive logistical exercise" was under way in Kent, with the surge in the number of migrants arriving set to continue.
In the last three months, the number of under-18 asylum seekers in the care of Kent County Council has nearly doubled to 605.
Mr Carter said: "Our social services are working all the hours that they possibly can and we have no more capacity to take many more in the coming weeks if the increase in numbers continues as in the past few weeks."
Mr Carter said the council faced a £5.5m shortfall in covering care costs and it was asking for help "from Theresa May down" to manage the crisis.
A national fostering agency is appealing for families to come forward to help cope with a five-fold rise in unaccompanied asylum-seeking children on its books.
Compass Fostering says it has received 275 referrals from local authorities in the past three months, compared with 56 for the same period last year.
The Local Government Association has urged the government to reimburse the costs councils face when unaccompanied child asylum seekers arrive in the UK.
Deputy Chairman Cllr David Simmonds said councils where children arrive are responsible for every aspect of caring, housing and educating them, through to the age of 25.
"The current situation is placing unprecedented pressure on an already overburdened system," said Cllr Simmonds
The prime minister has said the UK will not become a "safe haven" and warned that illegal immigrants would be removed if they reached the UK.
Speaking in Vietnam during his tour of South East Asia, Mr Cameron said: "Everything that can be done will be done to make sure our borders are secure and make sure that British holidaymakers are able to go on their holidays."
He said the situation was "very testing" because there was a "swarm of people coming across the Mediterranean, seeking a better life".
The Refugee Council attacked Mr Cameron's use of the word "swarm" as "irresponsible, dehumanising language".
Labour's acting leader Harriet Harman said the choice of words was "inflammatory", while Lib Dem leader Tim Farron described it as "deeply alarming" as the prime minister was talking about "some of the most desperate people in the world".
The last official estimates suggest there are about 3,000 migrants in Calais. It is not known how many migrants have reached Britain in recent months via the tunnel.
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