The children's bodies were found next to their father's in their family home.
Their two dogs were also found dead in the bungalow near the village of Gurteen, County Sligo, Ireland.
Toxicology tests are still being carried out, but it is believed Trevor Wallwork, his 12-year-old daughter Kim and nine-year-old son Harry died from carbon monoxide poisoning.
Mr Wallwork's wife Susan – for whom he is said to have been a full-time carer – is seriously ill in hospital where she is receiving chemotherapy and radiotherapy treatment for cancer.
The tragedy came to light on Sunday night when Mr Wallwork's step-daughter Vikki Whitehorne went to the remote, rural bungalow.
According to the Daily Mail, Miss Whitehorne, 22, is said to have tried to phone the house but became concerned when she couldn't get an answer.
She is believed to have warned them that she could smell gas where she found the family in the sitting room.
Mr Wallwork, 52, was still in his chair as if watching television. The children were lying on the floor near his feet. The set was still switched on, as were the lights on the Christmas tree, which the family had only recently decorated.
According to the report, a doctor who pronounced all three dead at around 11pm, although it was not immediately clear how long the bodies had lain undiscovered at the house.
Police are said to have discovered a cylinder of gas attached to a heater in the hallway near the sitting room.
The children's mother, Donna Wallwork, is disabled and has been living with her parents in Leigh, Lancashire since about 2008. Mr Wallwork is believed to have moved to the bungalow six years ago, when his marriage broke down.
"They were lovely.They seemed like really nice people. It's an awful terrible tragedy," one local said.
A police spokesman said: "It was a terrible tragedy."