Archie suffered “catastrophic” brain damage in an accident at home three months ago.
His parents, Hollie Dance and Paul Battersbee, have mounted an appeal bid after a High Court judge ruled that doctors could lawfully stop treatment.
The couple from Southend, Essex, insist their son's heart is still beating.
Sir Andrew McFarlane, Lady Justice King and Lord Justice Peter Jackson are listed to consider arguments at a Court of Appeal hearing in London on Thursday.
Archie Battersbee, 12
Hollie Dance
Mr Justice Hayden delivered a ruling on Friday after reviewing evidence at a hearing in the Family Division of the High Court.
He cited the "compelling and unanimous" medical evidence as he described the "bleak" picture to the "tragedy" of Archie's story.
Archie has not regained consciousness since April 7.
Doctors treating him at the Royal London Hospital in Whitechapel, east London, think he is “brain-stem dead” and say continued life support treatment is not in his best interests.
Archie's mum, Hollie Dance
James Manning
They are being supported by campaign group, the Christian Legal Centre.
Lawyers representing the Royal London Hospital’s governing trust, Barts Health NHS Trust, had asked for decisions about what medical moves are in Archie’s best interests.
Another High Court judge, Mrs Justice Arbuthnot, concluded that he is dead after initially considering the case in another hearing.
But Court of Appeal judges said the evidence should be reviewed, as they upheld a challenge by Archie’s parents against decisions taken by Mrs Justice Arbuthnot.
Mr Justice Hayden said evidence shows Archie suffered a “significant injury” to “multiple areas” of his brain and had not “regained awareness at any time”.
He said: “There is unfortunately no treatment possible to reverse the damage that has been caused to Archie’s brain.
“There can be no hope at all of recovery.”
The judge said he reached his conclusion with “profound regret”.
Andrea Williams, chief executive of the Christian Legal Centre, said: “This family have fought courageously to get to this point in taking a stand for Archie’s life.
“We continue to stand shoulder to shoulder with them as they appeal this ruling.”
