The parents of the terminally ill baby, Charlie Gard, have reportedly been in touch with the White House after Donald Trump intervened over his care.
The President tweeted that the US would be "delighted" to help the 10-month-old, who suffers from rare genetic condition mitochondrial depletion syndrome, which causes brain damage and muscle depletion.
A long legal battle has played out over the treatment of Charlie, after Great Ormond Street proposed turning off his life support.
His parents, Connie Yates and Chris Gard, challenged the decision but lost their battle when the European Court of Human Rights (ECHR) refused to intervene, effectively confirming that the machine could be turned off.
Doctors have agreed to give the parents a few more days with their son.
Pope Francis on Sunday called for Charlie's parents to be allowed to everything possible to treat their child.
A family spokesman said: "The White House has been in talks with Charlie's family, GOSH [Great Ormond Street Hospital], the UK Government, the Department of Health and the American doctor who wants to treat Charlie.
"President Trump has a very good understanding of the whole case and he did not make an off the cuff tweet."
Ms Yates added: "The support from the Pope and the President has given us hope. They are traditional men who believe in the family.
"They believe in our case and understand why we believe it is right to continue fighting so hard to save Charlie."
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