The 67-year-old, known for his appearances on The One Show and This Morning and his regular Just One Thing programme on Radio 4, was last seen as he set off hiking along St Nicholas beach at 1.30pm local time on Wednesday.
His wife, Dr Clare Bailey, alerted authorities after he failed to return by 7.30pm, but they were unable to locate him.
A post has been shared in a local Facebook group appealing to anyone on the island of about 2,500 people who may have seen Mosley.
It said: "A search and rescue team is coming from Athens with drones and other more sophisticated equipment to extend the search.
"Have you seen this man? He set off to walk back from St Nick's at about 13.30 and failed to make it home. His friends are concerned as it is 6 hours since they last saw him. His name is Dr Mike Mosley and he is a familiar face for many British people."
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