A Vatican pediatric hospital says experimental therapy "could have been an opportunity" to help Charlie Gard, but it was too late to start care for the critically ill baby.
Doctors at Great Ormond Street in London maintained that it was kinder for the Charlie's life support to be switched off.
Mr Justice Francis gave the parents and GOSH until noon today to reach agreement.
His parents would like up to a week to say goodbye to their little boy but his medics say death should come within a few hours.
Mr Justice Francis in April ruled in favour of Great Ormond Street and said Charlie should be allowed to die with dignity.
They said the couple now wanted a move to a hospice.
Barrister Grant Armstrong, who leads Charlie's parents' legal team, indicated that the couple wanted to privately fund treatment at a hospice where Charlie would stay for a number of days before life-support treatment was ended.
The child suffers from permanent brain damage and can not breathe on his own.
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Doctors at Great Ormond Street Hospital (GOSH) said it would not be practical for Charlie, who is severely brain damaged, unable to breathe without machines and can not move his arms and legs as a result of his mitochondrial DNA depletion syndrome, to receive treatment at their home and argued it is not in his best interest for him to be kept alive at a hospice for a long period of time.
Emerson said his daughter suffered from a 17-hour epilepsy fit, and afterward doctors told him that Jorja's case basically was hopeless.
It was previously determined Charlie will be moved to a hospice and taken off life support "shortly after" if no agreement was met.
"I became involved in Charlie's case when I was contacted by his parents, and I subsequently agreed to speak with his doctors to discuss whether an experimental therapy being developed in my lab could provide meaningful clinical improvement in Charlie's condition", Dr. Hirano said in a statement.
The judge issued reporting restrictions to prevent the public knowing when Charlie would die.
But the hospital said a paediatric intensive care unit specialist would be required to care for the child at a hospice over a longer period.
Monday, after months of proceedings, the parents of Charlie Gard had chose to renounce their court battle to preserve the lives of their children.
British courts, backed by the European Court of Human Rights, refused permission, saying the parents' plan was not in Charlie's best interests. His muscles deteriorating, Charlie has been kept alive on a ventilator in a London hospital.