An Iranian family have lost their appeal against a decision to remove their benefits and evict them from their home in Bury, Greater Manchester.
The Khanali family came to the UK claiming they were persecuted because they were Christians, but their application for asylum was refused.
The Home Office has the power to stop benefits to speed up their departure.
The two children could now be taken into care under section nine of the Asylum Act.
"It's a terrible situation the family are in and the council is now in a very difficult position," said councillor Tim Chamberlain, executive member for health and wellbeing at Bury Council.
"We've got to walk a very difficult tightrope between what we're required to do under the act and what we're obliged to do in terms of protecting the best interests of the children.
"As we understand it the best interests of the children are that they remain with the family.
"What we're now being asked to do under section nine of the Immigration Act undermines the Children's Act."
The family's solicitor, John Nicholson, said they were distraught at Tuesday's decision.
'Must return home'
"It's been an inhumane process all the way through section nine," he said.
"They've been hounded even before the birth of their six-month-old baby right the way through her early months and they don't know which way to turn.
"We are quite shocked because we think this law not only is inhumane but also that the Home Office have carried it out in a wrong way and we would ask the home secretary to reconsider the whole pilot scheme."
The family had won a similar hearing against the Home Office earlier in August, but now face having their benefits withdrawn unless they co-operate with the government and leave the UK voluntarily.
In a statement, the Home Office said it was sending "a clear signal...that failed asylum-seeking families who are refused permission to remain in the UK must return home".
The statement added: "Support will not be withdrawn where the family is co-operating with arrangements to return them to their home country and there is absolutely no reason for a family to become destitute or to be split up.
"We do not believe that many parents would put their children in this position."