Afusat Saliu, 31, and Bassy, four, and Rashidat, two, from Leeds, had been due to leave on a Virgin Atlantic flight to Lagos at 22:20 BST on Thursday.
The airline’s boss Sir Richard Branson received more than 1,000 tweets calling on him to intervene personally.
The Home Office would not confirm whether deportation had been delayed.
Ms Saliu and her daughters were arrested by UK Border Agency staff on Wednesday.
Anj Handa, a close friend of the family, said: “Afusat was meant to be given 72 hours’ notice before she was actually deported, and the Home Office totally ignored that. Finally they have accepted that.
“She is not going to be deported tonight or tomorrow morning but that does not mean she is not going to be deported.
“She has been given an overnight reprieve but the campaign is still very much ongoing and the lawyers are still working on her judicial review.
“We are still urging people to keep up the pressure to help her stay in this country while the case is reviewed.”
Ms Saliu’s solicitors BP Legal have launched a judicial review in an attempt to allow her to remain in the UK, reports the BBC.
They say officials have ignored their own guidelines by ordering her deportation before the review is heard.
Earlier, Sir Richard Branson said in a statement: “Without knowing the specifics on the case it is very difficult to comment in detail on a legal decision taken by the Home Office.
“While Virgin Atlantic cannot confirm whether any passengers are or are not flying with them due to data protection laws, they would view any case like this with the safety and welfare of all passengers their primary concern
“Regardless, if Afusat Saliu and her family are deported to Nigeria we call upon the UK and Nigerian governments to do all they can to protect the family and ensure they are not put at risk of female genital mutilation.”
Sir Richard described female genital mutilation (FGM) as an “horrendous practice and a serious violation of internationally recognised human rights”.
He called on the authorities to give the Saliu family all the support and protection possible.
Ms Saliu fled from Nigeria in 2011 after her stepmother threatened to subject Bassy to female genital mutilation. Ms Saliu, who is herself a victim of FGM, was heavily pregnant at the time.
She has said she fears both her daughters will be subjected to the practice and forced into marriage should they return to Nigeria.
The family converted to Christianity in the UK and worry they would be targeted by Islamist militants Boko Haram in the West African country.
It is understood the Salius 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.
A Home Office spokeswoman said it did not comment on individual cases.
Source: New Telegraph