The Holy Father, who met the father of Alfie last week in Rome, said he was "moved by the prayers and vast solidarity" shown for the toddler who is at the center of court battles between hospital authorities and parents to keep him alive.
Judges have approved plans for withdrawing treatment and bringing Alfie's life to an end.
Egan is the first English bishop to comment individually on the case, although the Bishops' Conference of England and Wales issued a statement on Wednesday affirming their "conviction that all those who are and have been taking the agonizing decisions regarding the care of Alfie Evans act with integrity and for Alfie's good as they see it".
Pope Francis said in a Twitter message today: "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".
Mr Diamond said there was now an "international relations element" to the case.
Mr Evans (pictured) posted on Facebook: 'Alfie has been granted Italian citizenship.
Hayden ordered Alfie's ventilator to be removed at 9.30pm local time but the family urgently appealed again to the Court of Appeal.
Archbishop Vincenzo Paglia, the President of the Pontifical Academy for Life, has called for a "shared response" to find a solution that could see terminally ill British toddler Alfie Evans receive treatment outside the United Kingdom.
The only slight hope of a reprieve is a last ditch appeal to the European Court of Human Rights by Alfie's parents. "Alfie belongs to Italy". "I love Alfie and I love Kate, I will not give up".More news: Netflix's The Witcher Series Gets New Information
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Hundreds of protestors have attempted to storm the hospital where Alfie Evans is on life support.
Two-year-old Alfie Evans suffers from an unidentified degenerative neurological condition and has been under continuous hospitalization since December 2016.
Specialists at Alder Hey said life support treatment should stop and Mr Justice Hayden said he accepted medical evidence which showed that further treatment was futile.
The Italian foreign ministry, in a brief note April 23, said Angelino Alfano, the foreign minister, and Marco Minniti, the interior minister, "granted Italian citizenship to little Alfie".
The couple then argued Alfie was being wrongly "detained" at Alder Hey and made a habeas corpus application. The call happened between the family's legal team, Justice Hayden, the Italian ambassador, and hospital administration.
The decision by the hospital to turn his life support off, against the will of his parents, has caused thousands of people to protest in person and on social media.
A spokesman for the European court said today: "The European Court of Human Rights has today rejected the application submitted by the family of Alfie Evans as inadmissible". Meanwhile pro-life demonstrators tried to get into the Alder Hey Hospital but were blocked by police, as tension mounted on Monday.
Early on Monday EHCR judges also refused to intervene.