Alfie Evans' parents wait for supreme court response
Alfie's parents now want to stage a fight at the Supreme Court in London.
The couple must first clear an initial legal hurdle and persuade Supreme Court justices that they have a case worth arguing.
A Supreme Court spokeswoman on Tuesday said justices had received a written application from the couple.
She said justices would probably decide whether the couple had a case worth arguing during the week starting Monday March 19.
The spokeswoman said justices might make that decision after considering written arguments or may stage a hearing where the merits of the couple's case could be debated.
She said three justices, including Supreme Court president Lady Hale, would consider arguments about the merits of the case and decide whether to give the couple permission to stage a fight.
Judges have heard that Alfie, who was born on May 9 2016, is in a "semi-vegetative state" and has a degenerative neurological condition doctors had not definitively diagnosed.
Mr Justice Hayden had analysed the case at hearings in the Family Division of the High Court in London and Liverpool.
Specialists at Alder Hey Children's Hospital in Liverpool said life support treatment could stop and the judge said he accepted medical evidence which showed that further treatment was futile.