Afusat Saliu, 31, and her daughters, Bassy, four, and Rashidat, two, from Leeds, were given a reprieve on 29 May.
But lawyer Bhumika Parmar said the family would now be deported "some time after 19:00 BST on Tuesday".
The Home Office said the case had gone through the "proper legal process".
Ms Saliu and her daughters were arrested by UK Border Agency staff on 28 May.
'Absolute mockery'
They been due to leave the UK on a Virgin Atlantic flight to Lagos at 22:20 BST last Thursday.
The airline's boss, Sir Richard Branson, subsequently received more than 1,000 tweets calling on him to intervene personally.
Their lawyer, Bhumika Parmar, said that earlier on Tuesday they were still being held at a detention centre near Gatwick airport.
She described their detention and imminent deportation as "an absolute mockery of our British government and protection".
A Home Office spokesperson said: "We consider every claim for asylum on its individual merits and in this case the claimant was not considered to be in need of protection.
"The case has gone through the proper legal process and our decision has been supported by the UK courts on five separate occasions, while the European Court of Human Rights declined an application to halt the removal."
Ms Saliu fled from Nigeria in 2011 claiming her stepmother had threatened to subject Bassy to female genital mutilation.
Ms Saliu, herself a victim of FGM, was heavily pregnant at the time.
She has said she fears both her daughters will be made to undergo the procedure and be forced into marriage should they return to Nigeria.
It is understood the Saliu family were taken into custody and detained at a holding centre in Sheffield before being moved to a family immigration centre in Crawley, West Sussex.
More than 120,000 people have signed a petition calling on the Home Office reconsider the case.
Source: BBC
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