Charlie was born in August with a rare genetic condition called infantile-onset encephalomyopathic mitochondrial DNA depletion syndrome, according to court records.
"The President is deeply moved by the heartbreaking situation facing Charlie Gard and his parents", a White House source told the newspaper.
On Tuesday, the parents lost a bid to take Charlie to the USA for trial therapy when the European Court of Human Rights sided with earlier rulings that continued treatment would cause "significant harm" and that life support should end.
May said that any parent would "want to do everything possible" for their child, but she declined to support a plea from parents Chris Gard and Connie Yates to allow the family to travel to the U.S.
The hospital where the baby is being treated, the Great Ormond Street Hospital in London, has asked for permission to stop the artificial ventilation of the child in order to provide palliative care.
But the Pope has now stepped in, offering Charlie treatment at the Vatican's own children's hospital in Rome.
British Foreign Secretary Boris Johnson told Italian Foreign Minister Angelino Alfano that it is "right that decisions continue to be led by expert medical opinion, supported by the courts", according to the Press Association in the United Kingdom.
However, it seems that the law has the final say in the fate of Charlie.
President Trump also weighed in on Twitter, telling Charlie's parents that the U.S.is ready to help any way we can.
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'.
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".
Charlie's mum Connie said: "The support from the Pope and the President has given us hope".
Enoc said she was contacted by the child's mother whom she describes as "a very determined and decisive person and doesn't want to be stopped by anything".