An American doctor offering to treat Charlie Gard might travel to London to examine the terminally-ill baby.
News of the possible visit emerged as Charlie's parents mounted the latest round of a legal battle at a hearing in the Family Division of the High Court in London.
Chris Gard and Connie Yates asked a judge to rule that 11-month-old Charlie, who suffers from a rare genetic condition and has brain damage, should be allowed to undergo a therapy trial in the United States.
Specialists at Great Ormond Street Hospital in London, where Charlie is being cared for, say the therapy is experimental and will not help.
They say life-support treatment should stop.
The couple, who are in their 30s and come from Bedfont, west London, have already lost battles in the High Court, Court of Appeal and Supreme Court in London.
They have also failed to persuade European Court of Human Rights judges to intervene.
The couple say there is new evidence and want Mr Justice Francis to carry out a fresh analysis of their case.
In April, the judge ruled in favour of Great Ormond Street and said Charlie should be allowed to die with dignity.
Mr Justice Francis began considering their claims at a hearing in the Family Division of the High Court in London on Thursday.
The hearing is due to resume on Friday.
The doctor gave evidence to the hearing on Thursday via a video link from the United States.
Mr Justice Francis has ruled that the doctor cannot be identified.
The judge said he wanted to hear what the doctor thought had changed since he gave his ruling in April.
The doctor said he had clinical data which was not available in April.
He said he still thought the therapy was ''worth trying''.
The doctor estimated a 10% chance of improvement in muscle strength and a "small but significant" improvement in brain function.
He said he had not seen or examined Charlie.
The judge said it may be a good idea if the doctor travelled to London to see Charlie and meet Great Ormond Street specialists.
Lawyers told the the judge that the doctor might be able to visit in the next few days.
Barrister Grant Armstrong, who is leading Charlie's parents' legal team, said the couple thought they had fresh evidence.
''The parents seek to re-open the case in relation to the chances of success of treatment,'' he said.
''This case does raise some important issues.''
Mr Armstrong told the judge in a written summary of the couple's case: ''The parents contend that there is a real issue to be resolved as to whether the court's earlier findings can be sustained.''
He added: ''The material shifts the balance of best interests clearly in favour of continuation of life and in favour of treatment.''
Mr Justice Francis said he was unlikely to give any ruling on Friday.
The couple's spokesman said outside the High Court that they had left to go back to their son's bedside.
Alasdair Seton-Marsden said:"Let's pray that British justice, known throughout the world for being fair, decent and compassionate, comes through for baby Charlie."
Mr Seton-Marsden said: "Chris and Connie, the parents of Charlie, apologise for not being able to be here in person.
"However, after a very long day in court they wish to return to the bedside of baby Charlie."
He added: "Today Charlie's case has moved from the chances of the ground-breaking therapy working from being as close to zero as to make no difference to, according to expert evidence heard today in court, being a small but significant chance."
Charlie's parents walked out of Thursday's hearing shortly before the lunch adjournment.
Mr Gard had stood up and said: "I thought this was supposed to be independent."
They returned for the afternoon session.