It's an effort to prevent doctors at Great Ormond Street Hospital in London from turning off his life support and to allow his parents to be the ones making the decision on whether to do so.
Mariella Enoc, president of the Vatican's Bambino Gesu Children's Hospital, released a statement asking the director of the London hospital, where Charlie is staying, "to verify whether the health conditions exist to possibly transfer Charlie to our hospital."
"We know that this is a desperate case and, apparently, there is no effective therapy," Enoc said.
Charlie's mother, Connie Yates, has been in contact with Enoc, Bambino Gesu's press office told CNN.
After that interaction, Enoc contacted the Great Ormond Street Hospital in London. Enoc said the London hospital was very kind and thanked her for her interest, but confirmed that for legal reasons, transferring Charlie to their care would be impossible.
The offer comes after an extensive legal battle over the fate of Charlie, a terminally ill infant who has a rare genetic disease called mitochondrial DNA depletion syndrome. It leads to weakened muscles and organ dysfunction, among other symptoms, with a poor prognosis
for most patients.
His parents, Chris Gard and Connie Yates, want Charlie to be released into their care so they can take him to the United States for an experimental treatment.
But last week, after a series of escalating legal appeals, the European Court of Human Rights ruled that the hospital can discontinue life support for the baby, who has been in the intensive care unit since October.
His desperate plight has come to the attention of Pope Francis and US President Donald Trump
"The domestic courts concluded that it would be lawful for the hospital to withdraw life-sustaining treatment because it was likely that Charlie would suffer significant harm if his present suffering was prolonged without any realistic prospect of improvement, and the experimental therapy would be of no effective benefit," the court said in a news release.
Life support was expected to be discontinued last Friday
The ruling to end Charlie's life support, against the parents' wishes, brought reactions from around the world.
On Sunday, the Pope called for the parents to "accompany and treat their child until the end."
"The Holy Father is following with affection and emotion the situation of little Charlie Gard and expresses his closeness to his parents. He is praying for them, in the hope that their desire to accompany and care for their own child until the end will be respected," the director of the Holy See Press Office, Greg Burke, said in a statement.
The Bambino Gesu Hospital statement began by quoting a tweet from Pope Francis on June 30: "To defend human life, above all when it is wounded by illness, is a duty of love that God entrusts to all."
The hospital has offered to keep Charlie on life support and allow his parents to decide "how to handle" ending his life, including when to switch off life support, it said.
"We are close to his parents with our prayers and, if they wish so, we are ready to welcome their child in our structure, for the time he has left to live," the hospital said.