The US reports also claim Cassidy's condition has deteriorated and the family are now "preparing for the end".
Cassidy is being treated at a Fort Lauderdale, Florida, area hospital, longtime publicist Jo-Ann Geffen told CNN on Saturday. "Geffen says there is nothing "imminent" about his condition, and doctors are hoping to 'keep him as well as they can until they can find another liver, '" ABC News reported on November 18.
The star was rushed to hospital with liver failure four days ago and was initially put in a medically induced coma for treatment.
David previously spoke out on his alcohol battle too, telling Piers Morgan in 2014: "If I take another drink, I'm going to die, physically, mentally, emotionally, spiritually".
Meanwhile, a source told U.S. website TMZ.com that the star's condition was "grim" and his family are being told they "need to prepare for the worst".
"When his brother Patrick walked into the room, David lit up like a Christmas tree", a source told the publication. There's been total resolution within the family.
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The two singers were introduced by host Jamie Foxx , who noted, "This year, we needed the power of music to help us escape the news of today ".
On Sunday evening, Beau Cassidy wrote a message on his father's official Facebook page.
In March, he wrote: "To the many thousands of people who have supported me over the last 5 decades I send my love". The site quoted the aforementioned statement from Cassidy's rep, adding that the actor and his family are taking things "day by day".
Earlier this year, David revealed that he has dementia - with the star forced to open up after fans grew concerned when he forgot his lyrics during a show.
"I was in denial, but a part of me always knew this was coming", said Cassidy, whose mother - Evelyn Ward - also suffering from the condition. "I was concerned people would mistake my alcoholism with my diagnosis", said Cassidy.
"I am enjoying some time at home, appreciating the warmth and contemplating what I will be doing next". I want to love. He was placed in the intensive care unit, and was diagnosed with organ failure.