The parents of a 23-month-old boy at the centre of a life-support treatment battle might ask the Supreme Court to consider their case for a second time.
Tom Evans, 21, and Kate James, 20, on Monday lost a second fight over Alfie Evans in the Court of Appeal.
A lawyer representing the couple had asked appeal judges to rule that Alfie should be allowed to travel to a foreign hospital.
Lord Justice Davis, Lady Justice King and Lord Justice Moylan ruled against them after an appeal court hearing in London.
Barrister Paul Diamond said the couple might make a further appeal to the Supreme Court.
The couple, from Liverpool, have already lost fights in the High Court, Court of Appeal,
Supreme Court and European Court of Human Rights.
In February, Mr Justice Hayden ruled that doctors at Alder Hey Children's Hospital in Liverpool could stop treating Alfie against the wishes of his parents following hearings in the Family Division of the High Court in London and Liverpool.
Specialists at Alder Hey said life-support treatment should stop and Mr Justice Hayden accepted medical evidence which showed further treatment was futile.
Alfie's parents say "the state" is wrongly interfering with their parental choice, and want to move Alfie to a hospital in Rome.
Mr Justice Hayden said flying Alfie to a foreign hospital would be wrong and pointless.
Court of Appeal judges upheld his decisions, and Supreme Court justices and European Court of Human Rights judges refused to intervene.
Last week, Mr Justice Hayden endorsed a detailed plan put forward by Alder Hey doctors for withdrawing life-support treatment, after considering a number of issues at a follow-up High Court hearing.
Mr Diamond on Monday asked the three appeal judges to overturn decisions made by Mr Justice Hayden last week.
Alfie's parents say their son has improved in recent weeks and had asked Mr Justice Hayden to allow a new assessment.
Mr Hayden refused that request. He said medical experts' unanimous view was that Alfie's brain had been eroded by disease and further assessment was pointless.
The parents also suggested Alfie was being unlawfully detained at Alder Hey, but the judge also dismissed that suggestion.
Appeal court judges dismissed the couple's challenge to Mr Justice Hayden's decisions.
Judges have heard that Alfie, born on May 9 2016, is in a semi-vegetative state and has a degenerative neurological condition doctors had not definitively diagnosed.