A senior figure in the Somali militant group al-Shabab has told the BBC it carried out a deadly attack on a shopping centre in neighbouring Kenya, BBC reported.
Al-Shabab, which has links to al-Qaeda, said the attack was in response to Kenya's presence in Somalia, where its troops have been fighting the militants since 2011.
A senior Kenyan security official has put the death toll at 11.
However, the Red Cross says at least 30 people were killed and dozens injured.
On its Twitter feed, al-Shabab, which has links to al-Qaeda, said it was behind what it called the "Westgate spectacle".
The attackers entered the Westgate centre at about 12:00 local time (09:00 GMT), throwing grenades and firing automatic weapons.
Dozens of shoppers fled; many were trapped inside. Officers have been going shop to shop to secure the area.
Al-Shabab also said on Twitter that its fighters were still battling Kenyan security forces inside the Westgate centre, some seven hours after the assault began.
One gunman was arrested, Kenyan officials said.
The permanent secretary in the Internal Security Ministry, Mutea Iringo, said special forces were "evacuating civilians and searching for the criminals".
Some witnesses said Muslims were told to leave and that non-Muslims would be targeted.
"They came and said: 'If you are Muslim, stand up. We've come to rescue you," said Elijah Lamau.
He said the Muslims left with their hands up, and then the gunmen shot two people.
The US state department says it has reports that American citizens were injured in what it called "a senseless act of violence".
The BBC's David Okwembah, in Nairobi, says this is one of the worst incidents in Kenya since the attack on the US embassy in August 1998.
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