The head of the Vatican's Bambino Gesu Pediatric Hospital said that the Italian defense ministry has a plane ready to transport Alfie to Italy if the courts allow it.
Alfie's father told reporters outside Alder Hey on Tuesday morning that his son had been breathing unassisted for nine hours.
Following the court hearing, a hospital spokesman said: "Our top priority therefore remains in ensuring Alfie receives the care he deserves to ensure his comfort, dignity and privacy are maintained throughout".
Tom Evans and Kate James took legal action after the High Court ruled Alder Hey Children's Hospital in Liverpool could end his care.
Speaking of what may happen at this hearing, Roger Kiska said: "My gut is telling me that Alder Hey will continue arguing that it's in Alfie's best interests to have life-sustaining measures terminated".
An emergency hearing is now scheduled in the high court today as the child's parents seek to try one more time to obtain permission to transfer Alfie to a hospital in Italy.
Kiska later said Alfie's parents' latest legal attempt followed their son's "miraculous survival" overnight.
Pope Francis again expressed support for the boy's parents tweeting on Monday evening: "Moved by the prayers and enormous solidarity shown little Alfie Evans, I renew my appeal that the suffering of his parents may be heard and that their desire to seek new forms of treatment may be granted".
"Medical specialists have said that Alfie is in a "semi-vegetative" state" and that further treatment would be "futile", adding that it would also be "unkind and inhumane". But Alfie's parents are concerned Alfie will die if he doesn't get care and possible experimental treatment in Italy. "Daily, Alfie moves, stretches, yawns, responding to tickling and noise and cuddling", a petition on Change.org states.More news: Yeddyurappa says son won't contest from Varuna, BJP workers protest
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According to the BBC, Justice Anthony Hayden ruled in February that Alfie's plight was "profoundly unfair", but agreed with Alder Hey that treatment measures would be futile, and that the almost two-year-old boy rather needs "good quality palliative care".
Roger Kiska, a solicitor for the Christian Legal Centre, who is supporting the parents Tom and Kate Evans, said: "It's been a miraculous evening to say the least".
However, his parents and their supporters raised millions of pounds and fought for him to be flown to America for an experimental treatment.
But a doctor treating Alfie, who can not be named for legal reasons and appeared in court in medical scrubs, said that would require a "sea change" in the attitude of Alfie's family.
Today, a British doctors group, The Medical Ethics Alliance, expressed its horror over the treatment of Alfie Evans that it called a "medical tyranny".
Justices at Britain's Supreme Court last week upheld a lower court's conclusion that it would be pointless to fly the boy to Rome for treatment.
Football fans at the Liverpool Roma match are being urged to take part in a touching tribute to Alfie Evans in the 23 minute.
Alfie has defied doctors' expectations to this point and his parents are headed back to court to fight for his life further.
The child, who had been born healthy by all indications, had been taken to the doctor prior to his admittance as he began exhibiting abnormal jerking movements.