"Paul was found by the SA Police Service, slumped behind the wheel of his vehicle on the main road of Alexandria, in the Eastern Cape. His vehicle was undamaged," Robin Thesen-Smith said in a statement.
She said her 48-year-old brother was found on Wednesday and had sustained an injury to his head.
"The cause of the injury is not yet known to us, but there is a police investigation underway."
Past operations
Thesen-Smith said her brother received his first donor heart at the age of 14 from a Xhosa man in 1980, and his second, three years later, from a girl who was fatally injured in a car accident.
She dismissed as "disinformation" and "nothing more than mischievous lies and the figment of an overactive imagination" a report in the Saturday Star that there was controversy about the first donor's race at the time of the operation.
Read:
After the heart transplant
Heart transplant: the risks
Racial classification
The newspaper quoted Paddy Chapple, who was apparently a friend of Thesen, as saying that Thesen lay on the operating table for four hours until the apartheid government decided that when a heart left the body it became neutral, and that Thesen's classification remained white.
"The fact that the first donor heart was one from a Xhosa man was of absolutely no relevance whatsoever. Our entire family was overjoyed to get a suitable donor, as was the transplant team," said Thesen-Smith.
Heart transplant pioneer Chris Barnard and his brother Marius Barnard performed the transplants on Thesen.
Health problems
In later years, he also had a by-pass operation and a pacemaker fitted.
She said despite his health problems, Thesen went on to compete in the Para-Olympics in Switzerland the 1980s.
More information on Transplants:
Heart transplants – an overview
Watch this video which showcases how a heart transplant is performed: