Militants attacked the Indian city on Wednesday
UK officials say there is no evidence yet that anyone involved in the attacks on Mumbai had links with Britian.
It follows reports on Indian channel NDTV that British citizens of Pakistani origin were among the militants.
But UK security sources told the BBC that India's authorities have said there is no indication so far that anyone shot or in custody is British.
Foreign Secretary David Miliband said it was "too early to say" whether any of the attackers were from the UK.
British officials have confirmed they are investigating reports of Britons being involved in the attacks, BBC security correspondent Gordon Corera said.
Sources have said they have asked the Indians for information through all available channels.
But the Indians have indicated that there is no evidence of British involvement.
Mr Miliband earlier said: "I'm afraid I can't tell you anything about the names or origins or sources of this attack at this stage.
"Obviously the Indian authorities are focusing on ending the incident before they are focusing on where it came from.
"I think it's right to say that at least one of the perpetrators is still alive and is being questioned by the Indian authorities, and obviously we'd want to follow that up as well.
"But it's too early to say where the people came from," he added.
Gordon Brown says ''everything possible'' was being done to help Britons in Mumbai
Prime Minister Gordon Brown said it was too soon to say whether Britons were involved, and Home Secretary Jacqui Smith said British authorities had "no knowledge" of any home-grown links.
One British national, Andreas Liveras, died and at least seven Britons were hurt in the attacks on the Indian city, which left at least 140 people dead.
The Foreign Office has issued an emergency number for people with relatives in Mumbai: 0207 008 0000.