Tania Clarence, 42, is being questioned by police after her three-year-old twin sons, Ben and Max, and four year-old daughter, Olivia, were found dead at their large home in New Malden, south west London, on Tuesday night.
Their father, Gary, who works as a director at City bank Investec, was reportedly away in the family's native South Africa with the couple's eight-year-old daughter when the deaths occurred.
A friend told the South African website Times Live that the three dead children were suffering from Spinal Muscular Atrophy (SMA) - a life-limiting genetic illness.
The inherited disease, which is also known as floppy baby syndrome, leaves children with little or no control of their movements.
The friend told Times Live: "They (Tania and Gary) did not go and have babies knowing they were giving them a genetic disease. She's been slammed for having three kids with SMA.
"They had the one child, their second child, who was young, only a few months old, when she fell pregnant (with the twins). It was not planned, but they felt they could cope.
"While she was pregnant, the second child was diagnosed as having SMA. The twins were then born premature and they stayed in hospital for a long time. The (parents) then asked for the twins to be tested for SMA as well.
"She came home with three kids severely disabled.
"Everything was regulated, everything they ate diarised. I bet Tania had not had a decent night's sleep in a few years."
The friend said the three children needed feeding tubes and, while the couple employed a carer and home help, they did not have round-the-clock assistance.
The friend added: "They tried to have a normal family life in the evening, by not having a house full of carers. Tania's borne the brunt (of it). Gary is a businessman and had to (travel).
"Everybody who met her could not but be compassionate. Most would think 'There but for the grace of God go I'.
"Everybody complains about their kids but at least they're healthy. It just puts your life into perspective."
A spokesman for the family told the Cape Argus that Mr Clarence was in "absolute shock" at the tragedy.
Lloyd Marshall also confirmed that the three children all suffered from the life-limiting genetic disease.
He said that, after the four-year-old was diagnosed with the condition, it was "50-50" as to whether the twins would also have it.
"Unfortunately, they did as well," he said.
He added: "Gary is in absolutely a state of shock - the whole family is. He never would have left the UK if he'd known (his wife) would be left really battling.
"Most of the family are on their way to the UK."
The family are still in mourning following the death of Mr Clarence's father, Brian, last October, the newspaper reported.
Described as a "pioneering entrepreneur", he founded the biggest hotel-conference centre in Southern Africa, the Birchwood Hotel and OR Tambo Conference Centre, which has 665 bedrooms and 60 conference rooms, the newspaper reported.
Mr Clarence's family, including his mother, Anne, and sister, Derri Phillips, are understood to have flown to the UK to support him.
Police said they are still holding Mrs Clarence, who cared full-time for her children at their home in Thetford Road, New Malden, at a police station.
Post-mortem examinations into the deaths of the three children will be held in due course, a Scotland Yard spokesman said.