Karl Andree, 74, was arrested in Jeddah in August last year after bottles of home-made wine were found in his car by police in the strict, alcohol-free kingdom, The Sun reported.
He is being held in the Red Sea city’s Briman Prison, which has a reputation for torture and overcrowded cells, as Saudi officials consider how to carry out the second part of his punishment – a public flogging.
The oil executive’s three children fear the punishment will kill him, as he is already weak from fighting cancer.
Simon told The Sun: "Our father has given 25 years of his working life to Saudi Arabia, and this is how he is treated. Until his arrest, he has always been happy working there and felt safe.
"He is 74 years of age, has had cancer 3 times and his wife is dying in a home in the UK.
"He now needs medical care for his cancer and Asthma, and there is no doubt in our mind that 350 lashes will kill him.
"We implore David Cameron to personally intervene and help get our father home. The Saudi Government will only listen to him."
A Foreign Office spokesman said: "Our embassy staff are continuing to assist Mr Andree, including regular visits to check on his welfare, and frequent contact with his lawyer and family.
"Ministers and senior officials have raised Mr Andree’s case with the Saudi Government and we are actively seeking his release as soon as possible."
