Paul Cantlie died on Thursday aged 80 from "complications following pneumonia", the family said in a statement.
A retired naval architect, Paul had earlier this month recorded a video plea from his hospital bed for IS to release his son.
"Mr Cantlie's physical strength had been waning since the kidnap of his younger son, John Cantlie, nearly two years ago," the family said in a statement.
"Many in the country will recall the recent broadcast from his hospital bed, when so demonstrably ill, but determined that his journalist son John should hear from his father 'how very proud I am of him'."
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It noted that efforts had been made to reach John Cantlie to tell him of his father's death before making it public.
The 43-year-old journalist has worked for publications including The Sunday Times and The Sunday Telegraph newspapers, as well as Agence France-Presse.