Charlie Gard's parents said they have been given more time before their baby son's life support is switched off.
Connie Yates and Chris Gard, from Bedfont, lost their final legal battle to take their son for experimental treatment in the US on Tuesday (June 27).
The parents wanted 10-month-old Charlie, who suffers from a rare genetic condition and has brain damage, to undergo a therapy trial in the US.
However specialists at Great Ormond Street Hospital (GOSH) said the therapy was experimental and would not help.
The latest stories in the case of Charlie Gard
The couple said they had been told Charlie's life support would be switched off on Friday but the hospital has confirmed it is working on plans for the baby's care, to give his family more time with him.
They have told Mail Online they have had talks with the hospital "and they have agreed to give us a little bit more time with Charlie".
Charlie's parents, both in their 30s, 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!"
A hospital spokeswoman said: "As with all of our patients, we are not able to and nor will we discuss these specific details of care.
"This is a very distressing situation for Charlie's parents and all the staff involved and our focus remains with them."
The spokeswoman added: "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."
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