Lawyers for the boy’s family took part in an hours-long hearing on Thursday, with the court in London sitting until late in the evening.
His loved ones have pledged to “fight to the end” with their last-minute bid to have him transferred to die in a hospice “with dignity”.
Doctors treating the schoolboy for the last four months declared Archie to be “brain-stem dead”, prompting a lengthy but ultimately failed legal battle by his family to continue his life support treatment in the hope he would recover.
The child has been in a coma since he was found unconscious at his home in Southend, Essex, on April 7 and is being kept alive by a combination of medical interventions, including ventilation and drug treatments, at the Royal London Hospital in Whitechapel, east London.
His mother, Hollie Dance, believes he was taking part in an online challenge at the time he became ill. He has not regained consciousness since.
Ms Dance said she wanted her son to “spend his last moments” together with family privately, complaining of a lack of privacy at the hospital.
She told Times Radio on Thursday: “We can’t even have the chance to be in a room together as a family without nurses.”
She added: “There’s absolutely no privacy, which is why, again, the courts keep going on about this dignified death – why aren’t we allowed to take our child to a hospice and spend his last moments, his last days together privately?
“Why is the hospital obstructing it?”
Barts Health NHS Trust has said Archie’s condition is too unstable for a transfer and that moving him by ambulance to a different setting “would most likely hasten the premature deterioration the family wish to avoid, even with full intensive care equipment and staff on the journey”.
A High Court order made in July requires that Archie remains at the Royal London Hospital while his treatment is withdrawn.
A family spokeswoman said a hospice has agreed to take him.