A spokesman at the Vatican's hospital said Wednesday that Great Ormond Street Hospital turned down the offer, citing legal reasons, but that officials are still working on a solution.
Trump tweeted his support, writing: '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 Donald Trump and Pope Francis brought worldwide attention to the family's legal battle this week by commenting on a case that pits the rights of parents to decide what's best for their children against the responsibility of authorities to make sure children receive the most appropriate care.
Charlie Gard suffers from a rare genetic condition and is unable to breathe unaided.
Shortly after his birth, Charlie was diagnosed with infantile-onset encephalomyopathic mitochondrial DNA depletion syndrome (MDDS).
After Pope Francis and President Donald Trump spoke out in support of Charlie, the Vatican's Bambino Gesu Children's Hospital asked whether the child could be transferred to Rome to receive care.
In a statement, the Vatican press office said Francis "is following with affection and sadness the case of little Charlie Gard and expresses his closeness to his parents". The parents of the toddler, who is now on life support in a London hospital, are fighting to take their son to the USA for an experimental treatment amid a court order rejecting their request.
Donald Trump "himself has not spoken to the family, he does not want to put pressure on them", said Ferré. "For this he prays that their wish to accompany and treat their child until the end is not neglected".
But the hospital said it had agreed to new plans with Gard and Yates "for his care, and to give them more time together as a family".
Charlie's mother has since told the Daily Mail that the support "has given us hope". In the post, the president claimed the United States would be "delighted" to help Charlie in any way possible. Specialists have said the proposed therapy wouldn't help Charlie.
"They believe in our case and understand why we believe it is right to continue fighting so hard to save Charlie".