A High Court judge has ruled that doctors can stop treating 21-month-old Alfie Evans, against the wishes of his parents, Kate James and Tom Evans.
Mr Justice Hayden, who analysed the case at hearings in the Family Division of the High Court in London and Liverpool, said he accepted medical evidence which showed further treatment was futile.
Specialists at Alder Hey Children's Hospital in Liverpool had asked Mr Justice Hayden to rule that life-support treatment could stop.
Alfie's parents, who are both in their 20s, have challenged Mr Justice Hayden's ruling and asked the Court of Appeal to consider the case.
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Three appeal judges analysed issues at a hearing in London on Thursday.
They said they aimed to deliver a ruling on Tuesday.
Barrister Stephen Knafler QC, who is leading Alfie's parents' legal team, told appeal judges that that "the State" had interfered with "parental choice".
He said Alfie's parents wanted to move him to a hospital in Europe and Mr Justice Hayden's ruling had prevented them from doing that.
"The State... has interfered with parental choice in a fundamental way," he said.
"Whether you regard the State as the hospital or the court.
"The parents have a choice which they can implement.
"The State, the hospital or the court, are prohibiting that."
He questioned whether Mr Justice Hayden had properly taken Alfie's parents' "vision" into account.
Mr Knafler said Alfie's parents would rather he died in an air ambulance than in Alder Hey.
"They would regard that death as Alfie continuing to fight for his life," he said.
"(They) would regard that death as noble and heroic, which is to be preferred to a managed death in a hospital setting."
He added: "They feel that Alfie's life has real, real value and importance."
Mr Knafler said there was a "profound" difference of opinion between Alder Hey doctors and Alfie's parents about what care was "appropriate".
"The parents' vision seems to have fallen somewhat by the wayside," he added.
"There is just a completely different vision here about what is best for Alfie."
Mr Evans said after the hearing that he was hopeful.
Judges have heard that Alfie, who was born on May 9 2016, was in a "semi-vegetative state" and had a degenerative neurological condition doctors had not definitively diagnosed.