Ms Kaplinsky was burned herself, while her daughter, Angelica, was so badly injured that she was flown back to Britain for emergency treatment.
Another family member, thought to be the girl’s grandfather, was less seriously hurt.
The family were rescued by local fishermen, who helped put out a fire that had spread from the engine room spread to the rest of the boat. He also towed them ashore.
The former BBC and ITN newsreader, who was the first winner of Strictly Come Dancing in 2004, owns a holiday home on the Greek island with her husband, Justin Bower.
"The girl suffered serious burns. Ms Kaplinsky and a man, thought to be the girl’s grandad, suffered lighter wounds," a source at the port told The Sun. “At first the girl was rushed to hospital in Corfu town with burns to her face and hands.
“Her mother and grandfather were treated at a private clinic. It was then decided to fly them home."
Angelica, Ms Kaplinsky’s youngest child, is said to be making a good recovery in hospital. Her burns are reported to be severe but not life-threatening.
The television presenter and her husband, both 45, were on the boat with Angelica, their 10-year-old son, Arlo and two other relatives when it caught light.
Corfu Port Authority told the paper: “Three people were injured after a fire on a boat. The vessel suffered a mechanical failure which caused a fire in the engine room, which was immediately extinguished by the people on the boat.”
A spokesperson for Ms Kaplinsky said: “Natasha and her family were involved in an accident in Corfu. They came home early to receive treatment and are making a good recovery at home.”
Investigators are examining the cause of the fire.
Ms Kaplinsky, who became Britain’s most highly paid newsreader in 2007 when Channel 5 recruited her from the BBC with a £1m-a-year deal, later went on to join ITV News as a presenter.
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