However, lawyers representing Great Ormond Street Hospital and a guardian appointed to independently represent Charlie's interests told Mr Justice Francis they were struggling to find any new evidence.
A British court will assess new evidence in the case of 11-month-old Charlie Gard as his mother pleaded with judges to allow the terminally ill infant to receive experimental treatment.
GOSH applied for another court hearing because of "new evidence relating to potential treatment for his condition".
But British and European courts have sided with the hospital's decision that the 11-month-old's life support should end, saying therapy would not help and would cause more suffering.
Parents of Charlie Gard, Connie Yates and Chris Gard pose for the media ahead of delivering a petition with more than 350,000 signatures to Great Ormond Street Hospital in London, Sunday, July 9, 2017.
Maxwell Smith, who lives in the United Kingdom with his parents, was diagnosed with mitochondrial depletion when he was only several months old, but at the age of five, he is alive thanks to nucleoside bypass treatment.
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His parents have claimed that they have evidence of treatment in the USA that could save his life, though Francis has given them by 2 p.m. on Wednesday to produce the details, with another hearing scheduled for Thursday.
There are 18 children currently on this treatment - one of them wasn't able to do anything and now she's riding a bike, she said.
A GoFundMe page set up to raise money for his treatment has so far raised more than £1.3m.
Screams from witnesses could be heard after the man - wearing a blue T-shirt and beige cap - raced across camera as BBC reporter Keith Doyle gave an update on Charlie Gard hearing on Monday. Connie Yates reportedly asked the judge.
Francis - who also ruled on an earlier chapter in the case - said everyone involved in the case wanted the best for Charlie.
"There are now seven doctors supporting us from all over the world - from Italy, from America, from England as well", Yates said Sunday when she arrived at the hospital.
Meanwhile, the Vatican has said the Pope is following the case "with affection and emotion" and "expresses his own closeness to his parents". -- This story has been corrected to show that the surname of the CEO of Americans United for Life is Foster, not Glenn.