Gard and Yates then filed a request with the European Court of Human Rights, an worldwide court based in Strasbourg, France, to consider the case, but the judges there also ruled in favor of his doctors.
Rare-disease specialists at Vatican-owned Rome children's hospital Bambino Gesù are working with other global experts to map out an experimental treatment protocol for Charlie, hospital chief Mariella Enoc said.
In a statement the Vatican said the Pope "expresses his closeness to his [Charlie's] parents".
Mr Trump tweeted on Monday: "If we can help little #CharlieGard, as per our friends in the United Kingdom and the Pope, we would be delighted to do so".
"President Trump has offered to help the family in this heart-breaking situation", Helen Ferre, White House director of media affairs, said in a statement.
Charlie's mother thanked the Vatican children's hospital for their offer, reiterating that "as long as he struggles, we struggle".
The Vatican-owned Bambino Gesù hospital has offered to take Charlie Gard, the severely ill 10-month-old baby whose parents are fighting to keep him on life support.
Who feels every pain he suffers more acutely than his own mother and father
However, the High Court in London ruled that the boy should not be transferred to the United States and that his life support should be shut down as he will be exposed to even more pain and suffering during the experimental treatment.
Reacting to the European court ruling, a spokesperson for the Catholic bishops' conference said the definitive ruling that "baby Charlie Gard can not undergo any further treatment is heartrending, most particularly for his parents and family".
Charlie's young life has become a cause celebre among American conservatives, who see it as a disturbing case of government overreach.
Charlie's parents have raised about $1.7 million to pay for Charlie's treatment and travel.
The pope's top bioethics official initially suggested that while the parents' wishes should be respected, they must also be helped to accept the limits of medicine. Last week, his parents lost a legal battle to take him to the US for trial therapy.
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