Members of Archie’s family want Court of Appeal judges to consider the case and are due to launch an appeal bid on Monday.
A spokesman for campaign organisation the Christian Legal Centre, supporting Archie’s family, said relatives will use a follow-up High Court hearing to ask Mrs Justice Arbuthnot to give them the go-ahead to mount an appeal.
The judge recently ruled that doctors could lawfully stop providing treatment to Archie after considering evidence at a trial in the Family Division of the High Court in London.
Relatives must establish they have an arguable case before a full appeal hearing can be staged.
Archie Battersbee is at the centre of a High Court life-treatment dispute
Hollie Dance
Archie Battersbee's mum, Hollie Dance, outside the High Court
James Manning
Doctors treating Archie at the Royal London Hospital in Whitechapel, east London, told the judge they think the youngster is “brain-stem dead”.
They said treatment should end and think Archie should be disconnected from a ventilator.
Archie’s parents, Hollie Dance and Paul Battersbee, from Southend, Essex, say the youngster’s heart is still beating and want treatment to continue.
Lawyers representing the Royal London Hospital’s governing trust, Barts Health NHS Trust, asked the judge to decide what moves are in Archie’s best interests.
Mrs Justice Arbuthnot heard that Archie suffered brain damage in an incident at home in early April.
Ms Dance said she found him unconscious with a ligature over his head on April 7.
The youngster has not regained consciousness.