Specialists at Great Ormond Street Hospital think it is time to stop providing life support treatment to eight-month-old Charlie Gard.
Doctors say Charlie, who suffers from a rare genetic condition and has brain damage, should move to a palliative care regime.
Charlie's parents, who are in their early 30s, disagree.
Postman Chris Gard and Connie Yates, from Bedfont, want to be allowed to take him to a hospital in the US for a treatment trial.
A judge has analysed the case at a hearing in the Family Division of the High Court.
Mr Justice Francis finished hearing evidence on Friday and is due to announce his decision on Tuesday.
The couple have launched an appeal on a GoFundMe website and have raised more than £1.2 million to pay for treatment after more than 80,000 people pledged money.
"Thank you to everyone who has supported us, you have put our faith back in humanity," they said in a statement posted on their GoFundMe page.
"We will always be eternally grateful whatever the outcome for Charlie."
They added: "We're praying that he gets his chance."
Mr Justice Francis has heard that Charlie, who was born on August 4, has a form of mitochondrial disease, a condition which causes progressive muscle weakness and brain damage.