Paula Parfitt, 41, of Strood Kent thinks Pippa Knight should leave hospital and wants specialists to stage a home-care trial.
Doctors treating Pippa, who is in a vegetative state at the Evelina Children’s Hospital in London, disagree and say life-support treatment should end.
Ms Parfitt has lost fights in the High Court and Court of Appeal.
Campaign group the Society for the Protection of Unborn Children (Spuc) has paid for lawyers to represent Ms Parfitt.
Officials say they have decided to fund a Supreme Court challenge.
They say lawyers are preparing an application to the Supreme Court.
A High Court judge ruled against Ms Parfitt earlier this year.
Mr Justice Poole decided that treatment could lawfully end and said Pippa should be allowed to die.
Three appeal judges last week upheld Mr Justice Poole’s decision.
Ms Parfitt said she was devastated and wanted Supreme Court justices to consider the case.
Spuc said on Tuesday that it would fund a Supreme Court application.
“Young Pippa’s life hangs in the balance and those of us who have seen the videos of her spending time with her mum can see how valuable that time is for her,” said John Deighan, Spuc’s deputy chief executive.
“We are willing to fight for the value of Pippa’s life no matter how ill she is.”
He said Pippa deserved the love and attention her family was desperate to give her and added: “Every human life is intrinsically valuable.”
Mr Justice Poole heard evidence at a trial in the Family Division of the High Court in London in December, and delivered a ruling in January.
The judge, who heard that Pippa’s father was dead, described the case as “heart-rending”.
Pippa was born in April 2015 and initially developed normally, but in December 2016 she became unwell and began to suffer seizures, the judge heard.
Doctors had diagnosed acute necrotising encephalopathy.