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It also reported four of the firm's Nepali tour guides are also missing.
The company's manager Kumar Adhikari said he was in contact with the guides until 08:40 when they were expected to be at Tibet's immigration office, but their phones were disconnected.
"We were hoping that inside it was safe," he told PA, "but later the video came from China's government which [showed] even Tibet's side was hit by the flood."
Adhikari said he was "slowly responding to" the families of the Britons but has been unable to give them any updates, adding he has been in contact with the British consulate and forwarded them copies of the clients' passports.
"I feel so sad. We are just thinking about what to do next...We are in pain at the moment, deep pain."
Other tour companies have also been impacted by the flash floods, with British citizens among other tours being affected.
The Trekkers' Society, which organises tours in Nepal, Tibet, and Bhutan, said nine UK nationals on a pilgrimage to Mount Kailash are missing after they crossed the Nepal-Tibet border.
It said four males, aged between 14 and 50, and five females aged 40 - 53 remained unaccounted for.
The society's general manager said the group were at Chinese immigration at the border when the flood happened, but it has no contact with the tour guide since.
Himalayan Glacier US told the BBC they have lost contact with a group of 47 guests and staff who where making a pilgrimage to Mount Kailash, which is considered the spiritual home of the Hindu god Shiva.
It said the group included two British citizens.
Kailash Journeys also reported a group of 27 guests and staff missing, with a spokesperson saying they were last in contact with them early Wednesday morning local time.
A list of names provided to the BBC included two men who are British citizens.
Fishtail Tours, a Nepali travel agency offering several climbing expeditions and tours of Nepal and Bhutan, said one British person was missing.
On Thursday, UK Foreign Secretary Ed Miliband said he had spoken to his Nepali counterpart, and said the country would do all it could to "help Nepal in its hour of need".
Miliband said he "offered financial, search and rescue and other help to aid the disaster response.
In a joint statement with Australia, the EU, Finland, France, Norway and Switzerland, the UK embassy in Nepal urged British nationals in Nepal to "exercise caution in affected areas".
