British people have been killed in the terrorist attack on a Kenyan shopping centre, the Foreign Office has confirmed.
Three Britons have been confirmed dead, and the number is likely to rise, officials said.
Prime Minister David Cameron said: "We should prepare ourselves for further bad news."
He condemned the "despicable attack of appalling brutality".
A total of 68 people have been killed after armed terrorists stormed the Nairobi shopping centre.
Earlier, Foreign Secretary William Hague said Britons had undoubtedly been caught up in the "callous and cowardly and brutal" assault by a heavily-armed gang who singled out non-Muslims.
Hostages were reportedly still being held as a standoff between security forces and the gunmen inside the Westgate mall continued.
Some of those shot were killed when they failed to recite a Muslim prayer at gunpoint, Sky News reported.
Kenyan President Uhuru Kenyatta vowed the terrorists would "not get away with their despicable and beastly acts".
And he revealed: "My nephew and his fiancee were among those who died in this attack."
Nobody should lose their life so needlessly and senselessly, he said.
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Terrified shoppers told of how they huddled in back hallways and prayed they would not be found by the Islamic extremist gunmen.
When the way appeared clear, crying mothers clutching small children and blood-splattered men sprinted out of the four-storey mall.
Witness Elijah Kamau said the gunmen told Muslims to stand up and leave and that non-Muslims would be targeted, as they began their attack.
The gunmen threw grenades and then opened fire, sending shoppers and staff fleeing for their lives.
Speaking from Nairobi, The Independent's Daniel Howden told the BBC that a "massive operation" was still ongoing
inside the mall this morning where the gang continued to hold an unknown number of people hostage.
Al-Shabaab, also known as Harakat al-Shabaab al-Mujahideen, earlier tweeted that there will be "no negotiations whatsoever".
The group claimed responsibility for the atrocity, writing on its official Twitter feed that "the Mujahideen entered Westgate Mall" at around noon local time yesterday.
It said it had previously warned the Kenyan government that, if they did not remove military forces from Somalia, there would be "severe consequences".
"For long we have waged war against the Kenyans in our land, now it's time to shift the battleground and take the war to their land," it tweeted.
It added that they were "still inside the mall, fighting the Kenyan kuffar inside their own turf" before the account was suspended by Twitter.
The organisation claimed it had killed more than 100 Kenyan "kuffar", a derogatory term used to describe non-Muslims.
Video footage of the attack showed terrified shoppers running for cover as the sound of gunfire could be heard.
The video, obtained by CCTV Africa, revealed the scenes in a supermarket in the mall, where some people tried to hide in the store's aisles as others fled for safety.
The footage showed shoppers abandoning their trolleys as repeated shots rang out.
At one point a group of people are seen hiding behind a door, peering through a small window as someone making a mobile phone call says "there's a terrorist".
Cameron said he had offered the Kenyan authorities "every assistance" in a call to president Uhuru Kenyatta.
He also indicated that British intelligence assistance could be provided.
Cameron said: "These appalling terrorist attacks that take place, where the perpetrators claim they do it in the name of a religion, they don't.
"They do it in the name of terror, violence and extremism and their warped view of the world.
"They don't represent Islam or Muslims in Britain or anywhere else in the world.
"Obviously the first priority is to make sure we do everything for the British nationals caught up in this, so we're strengthening the consular services that we have and the people that we have to look after our people.
"We've also offered the Kenyans help in terms of policing and advice and intelligence collaboration and other areas which we will be able to help with.
"The situation is ongoing and, as I say that why I think we have to brace ourselves for more bad news as this continues.
"It's an extremely difficult situation but we're doing everything we can to help the Kenyans in their hour of need."
Labour leader Ed Miliband offered his support to the Government and said: "Those who carried out this attack will be condemned across the globe. The cold-blooded killing of innocent women, children and men is as despicable as it is shocking."
An al-Shaabab spokesman told Channel 4 News that westerners would not be safe in Kenya until the country pulled its forces out of Somalia.
He said: "Kenyans have blood on their hands. Anyone who is prepared to come to Kenya must be prepared to face the reality, and we don't fear Europeans and Americans because we are not weak.
"And we are saying to the Europeans and the Americans who have been supporting those who have been attacking us, you should tell the Kenyans to stop their aggression if you want to be safe."
Kenya's Red Cross said the death toll from the militant attack on the mall had risen to 68 and president Kenyatta's nephew and his fiancee were among those killed.
"These are young lovely people I personally knew and loved," the president said, promising retribution against the terrorists.
"They shall not get away with their despicable and beastly acts.
"Like the cowardly perpetrators now cornered in the building, we will punish the masterminds swiftly and indeed very painfully.
"I call on Kenyans to stand courageous and united. Let us not sacrifice our values and dignity to appease cowards.
"Our victory must be conclusive - let us defeat them with our unity."
- A Foreign Office helpline has been set up for those worried about relatives in Kenya: 0207 008000