Doctors had been given permission to stop treating the 12-year-old, who has been unconscious for nearly four months, from Tuesday midday.
The family vowed to ask the Supreme Court for permission to appeal the ruling, which rejected a United Nations request to keep Archie on life support while one of its committees looked into the case.
The hospital said it would prepare to stop treating Archie but would not take any action while the family appealed.
Shortly after midday, the Supreme Court said it had received an application asking for permission to appeal.
It said Archie’s parents were seeking to delay when doctors can take their son off life support – which they have been granted permission to do by the High Court.
The family wan a delay to give the UN’s disability rights committee more time to look into whether removing Archie’s life support would be in breach of its convention.
The Court of Appeal turned down the UN’s request on Monday, saying the case was “very strongly in favour” of refusing to delay the removal of life support when considering Archie’s welfare.
“Every day that he continues to be given life-sustaining treatment is contrary to his best interests,” the lead judge said.
The appeal judges refused to grant permission to appeal against their ruling at the Supreme Court.
The family have now asked the UK’s highest court for permission to appeal, who said three Justices are looking into whether to allow this.
The Supreme Court said it will provide further updates when they become available.
Archie’s mother, Hollie Dance, said she was praying for an “encouraging response” from the court.
“We are having to battle over every decision with the hospital. There is nothing dignified in how we are being treated as a family in this situation,” she said.
“We do not understand what the rush is and why all of our wishes are being denied.”
Archie’s mother said she believes her son was “still with us” and was “progressing in so many ways”.
Doctors believe Archie is brain-stem dead and say continued life-support treatment is not in his best interests.
Archie suffered a brain injury in April during an incident at his home in Southend, Essex, which his mother believes may have been linked to an online challenge.
The 12-year-old boy has not regained conciousness since and has been kept in intensive care at the Royal London Hospital in Whitechapel.