Archie Battersbee's family will return to court on Thursday to demand the 12 year-old is transferred to a hospice, for what they're calling a "dignified" death.
However, doctors have claimed moving him from hospital involves "significant risk", and unless the High Court intervenes they plan to withdraw treatment on Thursday.
This comes after the European Court of Humans Rights (ECHR) rejected an urgent legal application on Wednesday evening.
In a letter to the family and their legal team, the letter said:
"We regret to inform you that on 3 August 2022 the President of the Court decided not to indicate to the Government of the United Kingdom, under Rule 39 of the Rules of Court, the interim measure you are seeking. Therefore the Court will not interfere with the decisions of the national courts to allow the withdrawal of life sustaining treatment from Archie Battersbee to proceed."
The Christian Legal Centre, which is supporting the family, said there have been offers from doctors in other countries to treat Archie.
Responding to the news, Archie's mum, Hollie Dance, said: "This is another heart-breaking development in our fight for Archie's right to live. We have been contacted by doctors in Japan and Italy who say they are willing to treat Archie, why can't we give him a chance?
"The NHS, the government and the Courts in this country and Europe may have given up on treating him, but we have not.
"The whole system has been stacked against us. Reform must now come through Charlie's Law so that no parents have to go through this.
"In a worst-case scenario, we want to take Archie to a hospice, but the hospital have said that we cannot do that despite previous promises. We have been told all along that this is all about Archie dying with 'dignity', and yet we are told we cannot take him to a hospice where it is quiet and we can spend time with him as a family without the chaos at the hospital.
"We will fight to the end for Archie's right to live."
On Wednesday morning, just before the 9am deadline set by Barts Health NHS Trust, the family's lawyers submitted an application to the ECHR based on a complaint about the UK's refusal to honour the UN interim measures which stated that Archie's life-support should not be removed.
The family said that this refusal failed to honour the UN CRPD Interim Measures and constituted a separate violation of international law, including the European Convention of Human Rights.
The UK courts have taken the view that UN CRPD Interim Measures have a different status in UK domestic law from ECHR Interim Measures, because ECHR rights are incorporated in domestic UK law whereas the terms of the relevant UN Convention are not so incorporated.
Andrea Williams, chief executive of the Christian Legal Centre, said:
"This is a story about a mother's love. She was catapulted into a whirlwind of legal battles and media attention all because she believed in giving her beloved son more time.
"Even though the whole system has come against her she has stood with her legal team against all the odds.
"The hospital and courts say this is all about Archie's dignity which means he should die quickly and at a specified time. The parents don't agree that it is dignified and in his best interests to die. Even the parents' request to move him to a hospice is being denied as undignified.
"Common humanity requires a different way of doing this.
"We continue to stand with the family as they consider their next options."