The number of young asylum-seekers who have been placed in foster care after arriving at the port town stood at 238 a year ago, rising to 369 by April, but has now gone up to 605.
Kent County Council Leader Paul Carter said the authority is looking into the possibility of sending the youngsters elsewhere in the country.
The county has also called on the Home Office for help to cover the estimated annual £5.5m cost of looking after asylum-seeking children.
“It is putting enormous strain on our children’s social services,” he said. “We are now caring for 605 under-18s and face a shortfall of £5.5m in costs to care for them.
“Staff are working flat out to support these vulnerable young people through our reception centres and we are urgently looking at new premises to expand the facilities.
“We are now working with the ADCS to come to a dispersal arrangement outside of Kent as we need places across the country where they can go after assessment.”
Alison O’Sullivan, president of the ADCS, said the situation in Kent has become "critical".
“There has been a gradual increase in numbers of unaccompanied asylum-seeking children in the past few months and a spike in July has created an acute and immediate capacity issue for the council,” she said.
“These children, some as young as 12, are extremely vulnerable and in need of our help and support. Our sole focus needs to be on the best interests of the children and ensuring that they are properly supported and cared for.”
She said some directors of children's services in the region are already providing accommodation to some of the vulnerable children.
But she said more help is required.
"Because of the unprecedented pressure on London and the South East, we are also asking authorities from across the country if they can also help," she said.
"And in the longer term, we are working with various government departments including the Department for Education and the Home Office on a sustainable national response.”
Earlier this month, it was reported that Kent County Council is planning to house asylum-seekers in a disused care home.
The number of asylum-seeking children currently being looked after by Kent County Council represents about a third of the total number of looked-after children in the authority.
As of the end of March 2015, there were 1,870 looked-after children in the authority.
A Home Office spokesman said: “Local authorities are responsible for providing suitable care to unaccompanied asylum-seeking children.
"Together with the Department for Education, we are working closely with Kent County Council to discuss the options available to them."