Explosions and heavy gunfire were reported earlier as soldiers stormed the Westgate shopping complex.
Three “terrorists” were killed and soldiers are continuing to comb the building floor by floor “looking for anyone left behind”, officials said.
The Kenyan Red Cross has told the BBC that 63 people remain unaccounted for.
The official death toll stands at 62 and more than 170 have been injured. There are fears the death toll will rise further.
British Defence Secretary Philip Hammond said that his best estimate was that six Britons had been killed.
The Somali Islamist al-Shabab movement has said it carried out the attack in retaliation for Kenyan military operations in Somalia.
“The terrorists could be running and hiding in some stores, but all floors now are under our control,” Kenyan Interior Minister Joseph Ole Lenku said.
“There is no room for escape.”
He told the BBC late on Monday that the operation would continue overnight, but stressed it was in its final stages.
The minister added that “it is very unlikely that there are any hostages” left in the complex.
The BBC’s Gabriel Gatehouse, who is at the scene, says there is still no clear picture of what is going on inside; officials have said forces are in control of the mall but have not said if they have arrested or killed all attackers.
As night fell on Monday, flames and thick smoke continued to rise from the complex.
The Kenya Defence Forces said the fire had been started by “terrorists to distract the ongoing operation”, and that the blaze was being managed by firefighters.
President Barack Obama said the US was offering its full support to the Kenyan authorities.
Kenya is one of the largest recipients of US security assistance in sub-Saharan Africa, with much of that aid focused on counter-terrorism.
The BBC’s defence correspondent Jonathan Beale says the US is likely to have responded to the crisis by sending one of its Foreign Emergency Support Teams (Fest).
But there was no suggestion they are directly involved, our correspondent says.
The KDF said 10 bodies had been retrieved from the building in the last 24 hours. More than 200 civilians have been rescued, 65 of whom remain in hospital.
Eleven KDF soldiers were injured during the operation, it said.
Earlier, police used tear gas to disperse crowds of onlookers gathered close to the Westgate Centre.
The Interior Ministry has been issuing regular warnings for people to stay away for their own safety.
Security has also been stepped up at entrance and exit points across the country, with “more than 10 individuals” arrested in relation to the attack, the ministry said.
It did not specify when or where the arrests were made, but media reports from Kenya said the suspects were detained at the Jomo Kenyatta International Airport.
More than 1,000 people were inside the mall complex when the attack began on Saturday.
Dr Sunil Sachdeva, a dentist who runs a clinic inside the mall, described the scene as the attack unfolded.
“There was a tent where a cookery competition for children was carrying on and there were bodies lying under there,” he told the BBC.
“There’s a very famous radio presenter in Kenya, she was shot. The scene was carnage and there was a guy lying right in the corner. He was cut to shreds.”
Prominent Ghanaian poet Kofi Awoonor – who was attending a literary festival in Nairobi – also died, as did a Chinese woman.
French, Dutch, South African, Indian and Canadian nationals are also among the foreigners confirmed killed, along with a dual Australian-British national.
Thousands of Kenyans have been responding to appeals for blood donations.
Al-Shabab, which is part of the al-Qaeda network, has repeatedly threatened attacks on Kenyan soil if Nairobi did not pull its troops out of Somalia.
There are about 4,000 Kenyan troops in the south of Somalia, where they have been fighting the militants since 2011 as part of an African Union force supporting Somali government forces.
Al-Shabab is fighting to create an Islamic state in Somalia.
Despite being pushed out of key cities in the past two years, it still remains in control of smaller towns and large swathes of the countryside.