The BBC's Gabriel Gatehouse: "It is in this supermarket that at least some of the attackers are now believed to be holed up"
Heavy gunfire and explosions have been heard at the shopping centre in Nairobi where militants are said to be holding a number of hostages.
Sixty-nine people have been killed and more than 170 injured since the attack began in Kenya's capital on Saturday.
Between 10 and 15 attackers are still inside the Westgate shopping centre. The Somali al-Shabab movement has said it was behind the attack.
Reporters at the scene said there had been heavy and rapid bursts of fire.
The International Criminal Court (ICC) announced on Monday morning it was adjourning the trial of Kenya's Deputy President William Ruto on charges relating to violence following elections in 2007.
The court said Mr Ruto would be excused from the trial, which began earlier this month, for a week to return to Nairobi to help deal with the crisis.
'Speedy conclusion'
The BBC's Mike Wooldridge, in Kenya's capital, says that the battle to end the long stand-off is continuing in earnest as conditions for those trapped inside the complex deteriorate.
An unnamed Kenyan security source told the AFP news agency early on Monday morning that an army assault was underway.
An AFP correspondent at the scene said he heard about 15 minutes of fierce gunfire which then subsided.
A photographer accompanying the correspondent said troops deployed around the mall were forced to duck for cover.
The photographer said it "sounded as if the shots were coming from somewhere around the mall, or were being fired from a vantage point in the mall".
The defence forces said on Twitter several hours ago: "All efforts are under way to bring this matter to a speedy conclusion."
Authorities have emphasised that it is a delicate operation as the safe rescue of the hostages is their top priority, says our correspondent.
Four soldiers had been injured in the attack by Sunday evening.
Kenyan military vehicles have been seen entering and leaving the area throughout Monday morning.
A Kenyan security official told the BBC that Kenyan forces were receiving foreign assistance, namely from Americans and Israelis.
'More bodies'
Overnight reports said that the gunmen were holed up in a supermarket.
Earlier, defence spokesman Col Cyrus Oguna said only a small number of hostages were still being held and most had been rescued.
As troops continued to clear the building, it was possible they would come across more bodies, Col Oguna warned on Sunday.
Troops at the mall complex were forced to take cover amid fresh firing on Monday morning
Security forces have imposed a cordon around the mall
More than 170 people have been injured over the course of the attack
In a news conference on Sunday, Kenyan President Uhuru Kenyatta said the country was united and strong in adversity.
"The criminals are now located in one place within the building," he said.
"With the professionals on site, we have as good a chance to neutralise the terrorists as we could hope for," he said.
Mr Kenyatta said his nephew and the man's fiancee were among the dead.
The UK Foreign Office has confirmed that three Britons have been killed, and says the number is likely to rise.
British Prime Minister David Cameron said in a tweet on Monday morning that he was returning to London to chair a meeting of the UK's emergency response committee, Cobra.
Mr Kenyatta's deputy, Mr Ruto, is due to return to Kenya on a morning flight from the Netherlands after being excused from the ICC trial.
Mr Ruto, an ethnic Kalenjin, is accused of planning to set up militias to attack the Kikuyu ethnic group in violence following a disputed presidential poll in 2007 which saw then-President Mwai Kibaki, a Kikuyu, win a second term in office.
Mr Kenyatta also stands accused of developing plans to attack members of other ethnic groups and is himself due to stand trial in November.
Repeated threats
More than 1,000 people were inside the mall complex when the attack began on Saturday.
The wife of an American working for the US Agency for International Development was killed, US officials said.
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 responded on Sunday to appeals for blood donations.
Al-Shabab says it carried out the attack in response to Kenyan military operations in Somalia.
The group, 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.
