Ms Kaplinsky and her father were said to have been lightly injured, but the child was so badly affected that she had to be flown back to Britain for treatment.
The newspaper reported that the girl’s condition is not thought to be life-threatening but she remains in hospital.
A spokesman for the Corfu Port Authority said: “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 spokesman for Ms Kaplinsky told the newspaper: “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.”
Ms Kaplinsky later posted a photo on Instagram of a bouquet of flowers with the caption: "Thank you everyone for your kindness."
Ms Kaplinsky, who has presented news programmes for the BBC, Channel 5 and ITN, was last year made an OBE for her services to Holocaust commemoration.
Over 15 months, the television star, whose paternal grandparents migrated from Poland to South Africa in 1929, interviewed 112 Holocaust survivors and concentration camp liberators as part of a commemoration project for the UK Holocaust Memorial Foundation.
