A British court on Monday gave the parents of 11-month-old Charlie Gard a chance to present fresh evidence that their terminally ill son should receive experimental treatment.
It ruled the children's hospital could turn off his life support.
Judge Nicholas Francis said a new court date has been set for Thursday.
She said they had received support from doctors from all over the world, as well as the UK.
"The Holy Father follows with affection and emotion the story of Charlie Gard and expresses his own closeness to his parents", read a July 2 statement issued by Vatican spokesman Greg Burke.
In a joint statement, they said: "Our bill will support Charlie's parents" right to choose what is best for their son, by making Charlie a lawful permanent resident in the United States in order for him to receive treatments that could save his life.
On Friday, Great Ormond Street Hospital for Children, where Charlie is being cared for, said it wanted the court to hear new evidence relating to the case, "in light of fresh evidence concerning potential treatment".
The interventions of the pope and the USA president have been the biggest help in keeping 11-month-old Charlie Gard alive, the critically ill baby's mother has said.
Anger Parents of Charlie Gard Connie Yates and Chris Gard leave the High Court with a spokesman
Charlie Gard suffers from a rare genetic condition and his doctors say that there is nothing that can be done to save him and he should be allowed to die with dignity.
That ruling by Justice Francis, who forbade the hospital from transferring Charlie to any other facility to receive so-called "nucleoside" therapy, was upheld on appeal and then by Britain's Supreme Court.
He said: "It seems that if he remains at Great Ormond Street it seems that nearly certainly his life support will be withdrawn and clearly he is likely then to die".
The case of baby Charlie Gard has the gravest consequences for the rights of parents and the autonomy of the family.
But, after a worldwide media storm over the case, global experts presented Charlie's family with new evidence that gave the baby a much higher chance of survival than previously thought.
"There are now seven doctors supporting us from Italy, America, and England, as well, that think that this [treatment] has a chance", Connie Yates said.
Charlie's rare mitochondrial disease has left him blind and deaf - unable to move his limbs. The hospital contends Charlie is brain-damaged and beyond medical hope, and the hospital wants to shut off his life support. He stressed that while the therapy was a treatment, not a cure, his son was "getting stronger every day". The hospital also offered to send the drug to Great Ormond Street Hospital - the British hospital where Charlie is being treated - if approved, the Post added.
Thanking supporters, Mr Gard said: "Let's get Charlie the treatment he needs".
Trump and Pope give baby's parents 'hope'
He suffers from mitochondrial depletion syndrome, a rare genetic illness which leads to progressive brain and muscle damage. President Trump also weighed in on Twitter, telling Charlie's parents that the U.S.is ready to help any way we can.