The Bee Gees star has been receiving hospital treatment after falling into a comatose state while battling colon and liver cancer and pneumonia.
Speaking to the Sunday Express, Robin-John revealed that Robin had addressed him, saying "Hello RJ" after becoming conscious for the first time in almost two weeks.
He also revealed that the singer is hoping to recover in time to attend the unveiling of the Bomber Command Memorial in June, which honours the 55,000 aircrew who died in World War Two.
Robin-John told the newspaper: "Dad, thank heavens, has awoken from a 12-day coma.
"He will do everything in his power to build himself up so he may attend the unveiling of the memorial, of which it was his greatest honour to be a part."
He added: "Being at the unveiling is something my father would never miss.
"My father holds this to be one of his crowning achievements and views as sacred his involvement in this project."
On Friday (April 20), it was reported that Robin's brother and fellow Bee Gees star Barry Gibb was singing at his hospital bed, where Robin-John, his mother Dwina and siblings Melissa and Spencer were maintaining a vigil.
Listen to Robin Gibb's song 'Don't Cry Alone' below:
> Robin Gibb 'made list of ambitions before coma'
> Robin Gibb: 'Cancer is payback for Bee Gees success'