Children from the Jungle camp in Calais who have relatives in the UK are expected to arrive in the country as early as next week.
British officials were sent to help French authorities speed up the removal of the youngsters to the UK before the site is demolished.
Under the so-called Dublin regulations, asylum claims must be made in the first safe country a person reaches - but children can have their application transferred to another country if they have family living there.
A Home Office spokesman said: "Work is continuing on both sides of the Channel to ensure this happens as a matter of urgency."
He added that Home Secretary Amber Rudd had made it "crystal clear" to the French Interior Minister that she intends to "transfer as many minors as possible" who are eligible under the rules.
Campaigners say they have identified hundreds who have a right to come to Britain - either because they have family ties here or through a programme to give refuge to unaccompanied minors from Europe.
The sense of urgency surrounding the issue was reinforced by a warning that there could be a "mass disappearance" unless the transfer of children is secured before the closure of the camp.
A report by Tomas Bocek, the Council of Europe secretary-general's special representative on migration and refugees, said the threat of imminent eviction was causing children to put their lives at risk to get to Britain.
It said: "A mass disappearance, like the one that occurred during the eviction of the southern part of the Jungle, is considered highly likely unless the transfer of eligible children to the United Kingdom is secured before the closure of the camp."
Stephen Hale, chief executive of the charity Refugee Action, said: "It's fantastic news that, at last, vulnerable children in Calais will be reunited with their families in Britain.
"These children must have the support they need to rebuild their lives when they arrive."
He added that town halls "also require long-term funding arrangements from government so that the commitment to support those children starting a new life in the UK is properly funded".
Ms Rudd told the Commons last Monday that more than 80 unaccompanied children had been accepted for transfer under the Dublin regulations so far this year.
French President Francois Hollande has said the site will be cleared by the end of the year.