Antony, a renowned clinical psychologist, is believed to have collapsed at the Tanglin Club, where he was staying.
He was here to pay a visit to Nicole's sister Antonia Kidman and was staying at the club as his son-in-law is a member there.
"Police were informed today at about 0954 hours [local time] that a man in his 80s was pronounced dead at Tan Tock Seng Hospital," a Singapore police spokesman said.
"Police are investigating the unnatural death," he was quoted by the Straits Times as saying.
Antony, who was close to both his daughters, was a clinical psychologist at Royal North Shore Hospital and a director of health psychology at the University of Technology Sydney.
Antony was the Director of the UTS Health Psychology Unit at Sydney's Royal North Shore Hospital.
He also ran community service seminars on aspects of mental health, was a regular speaker on radio and television and wrote for academic journals and the general public.
He was awarded an Order of Australia in 2005 for his contributions to clinical psychology.
No official statement from his family has been released yet.
