The balloon was flying at 1,000 ft (300 m) when it caught fire and exploded, plunging onto fields west of Luxor, officials said.
One witness said people were jumping out of the balloon, “from about the height of a seven-storey building”.
Two people, including the balloon’s pilot, reportedly survived the crash.
Luxor lies on the banks of the River Nile and is home to some of Egypt’s most famous pharaonic-era ruins.
The crash happened on one of the many dawn hot air balloon flights that give tourists an aerial view of Luxor’s famous sites, such as Karnak temple and the royal tombs in the Valley of the Kings.
It was understood that a gas cylinder exploded on board the balloon, bringing it down in an agricultural area just outside Luxor.
Cherry Tohamy’s balloon was landing when she heard an explosion and saw flames from a balloon above.
“Our pilot told us that the balloon had hit a high pressure electrical cable and a cylinder on board exploded,” said Cherry Tohamy, an Egyptian living in Kuwait who was on holiday in Luxor.
“People were jumping out of the balloon from about the height of a seven-storey building.”
She said ambulances were at the scene within 15 minutes.
Another witness, US photographer Christopher Michel said his balloon was just about to land when he “heard an explosion and saw smoke”.
NBC News quoted a Luxor health official as saying that the victims include nine from Hong Kong, four from Japan, two from the United Kingdom and two from France, with two yet to be identified.
The British foreign office said it was making urgent inquiries with its colleagues in Egypt to confirm reports of British casualties.
A spokesman from the Hong Kong government confirmed that nine Hong Kong residents were in the crash balloon and a team of immigration officers was being sent to Egypt.
Hot air balloon crashes have happened in Luxor before. Two British women were among 16 injured when their balloon came down in April 2009.
Luxor, like many other parts of Egypt, has seen a sharp downturn in visitor numbers since the uprising in early 2011 that forced long-time President Hosni Mubarak to step down.
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