Chris Gard and Connie Yates wanted 10-month-old Charlie, who suffers from a rare genetic condition and has brain damage, to undergo a therapy trial in the US.
But specialists at Great Ormond Street Hospital (GOSH) said the therapy was experimental and would not help.
The couple said they had been told Charlie's life support would be switched off on Friday, but the hospital has confirmed they are working on plans for the baby's care and to give his family more time with him.
A hospital spokeswoman said: "Together with Charlie's parents we are putting plans in place for his care, and to give them more time together as a family.
"We would ask you to give the family and our staff some space and privacy at this distressing time."
Charlie's parents, both in their 30s and from Bedfont, west London, had asked European court judges in Strasbourg, France, to consider their claim after judges in the High Court, Court of Appeal and Supreme Court in London ruled in favour of GOSH doctors.
But on Tuesday the European Court of Human Rights refused to intervene.
The couple say they have been denied their final wish to be able to take their son home to die and feel "let down" following the lengthy legal battle.
Alongside a video posted on YouTube on Thursday, Charlie's parents wrote: "We are utterly heartbroken spending our last precious hours with our baby boy.
"We're not allowed to choose if our son lives and we're not allowed to choose when or where Charlie dies.
"We, and most importantly Charlie, have been massively let down throughout this whole process."
Charlie's plight has touched many people and the family received donations totalling more than £1.3 million to take him to the US for therapy.
Campaigners also pledged their support to the family on social media with hashtags including #charliesfight #Fight4Charlie and #LetCharlieGoHome.
His parents said Charlie would die "knowing that he was loved by thousands", adding "thank you to everyone for all your support!"