Parents Tom Evans and Kate James have lost a series of legal challenges to continue treatment for the 23-month-old, who has a degenerative neurological condition.
The couple from Liverpool want to move their son from Alder Hey Children’s Hospital to Rome where they say doctors are willing to treat the little boy.
High Court judge, Mr Justice Hayden, will oversee a further hearing on the case in Manchester at around 3.30pm.
Speaking outside Alder Hey, Mr Evans said doctors were “gobsmacked” that his son was breathing nine hours after his life support ended.
“He’s still breathing now,” Mr Evans said.
“It’s come to a point when his mum’s actually asleep next to him so she can go to sleep, she feels comfortable with him.”
Mr Evans said it became obvious he was breathing “within a few minutes” of life-support being withdrawn on Monday night, although doctors re-intervened after he asked them to help.
He added: “Because he’s been doing it for nine hours totally unexpected, the doctors are gobsmacked and I do believe he will need some form of life support in the next couple of hours and I think he ought to be respected and given that.”
Alder Hey said it would not be issuing updates on Alfie’s condition out of respect for his privacy.
The children’s hospital also urged people not to contact its staff or switchboard seeking news about the youngster.
Alfie has been granted Italian citizenship in a bid to secure his transfer to Rome and last week Mr Evans met Pope Francis, who has voiced support for the couple.
In February, Mr Justice Hayden ruled that doctors at Alder Hey could stop treating the youngster against the wishes of his parents.
The judge said he accepted medical evidence which showed that further treatment was futile.
Court of Appeal judges upheld his decision, while Supreme Court justices and European Court of Human Rights judges refused to intervene.
Hundreds of Alfie’s supporters gathered outside Alder Hey on Monday as police were forced to block dozens from trying to storm the hospital building.