Connie Yates's and Chris Gard's son Charlie is receiving 24-hour treatment for a condition that causes progressive muscle weakness.
UK doctors have said there is no accepted cure and Charlie should be allowed to die with dignity.
A High Court judge will decide on Monday what is in his best interests.
The judge will analyse medical evidence at a hearing in the Family Division as Charlie's parents seek to keep him on life support at London's Great Ormond Street hospital.
Though born healthy in August 2016, Charlie began to lose weight and strength by the time he was eight weeks old.
He was admitted to Great Ormond Street in October after developing aspiration pneumonia and was diagnosed with mitochondrial depletion syndrome - a condition that causes progressive muscle weakness.
Last month, his parents told the BBC's Victoria Derbyshire programme that they would continue to fight for their child's survival.
They said: "He can move his mouth, he can move his hands. He can't open them fully, but he can still open his eyes and see us, in response to us.
"We don't feel he's in pain at all."
The couple, who live in London, believe Charlie may have a chance of surviving if he receives pioneering treatment in the US.
"We just want to have our chance. It would never be a cure but it could help him live. If it saves him, amazing," Miss Yates said.
"I want to save others. Even if Charlie doesn't make it through this, I don't ever want another mum and their child to go through this."
Charlie is thought to be just one of 16 children in the world to suffer from mitochondrial depletion syndrome.
Earlier this month, a spokesperson for Great Ormond Street Hospital for Children NHS Foundation Trust said: "Charlie has a very rare and complex disease, for which there is no accepted cure.
"Charlie was very unwell when he was admitted to Great Ormond Street Hospital and has remained under 24-hour care on our intensive care unit.
"But his condition has continued to deteriorate and we now feel we have exhausted all available proven treatment options.
"We cannot imagine how hugely distressing this is for his family.
"We continue to support them in every way we can, while advocating, what we believe, is best for Charlie."
The parents of baby with rare genetic condition have reached a £1.2m crowdfunding target for him to have pioneering treatment in the US.