A senior police source said early on Tuesday the operation was “over”, however journalists at the scene have reported sporadic gunfire at the mall.
There has been no official confirmation the siege is over. Kenya’s president is expected to make a statement shortly.
At least 65 people have been killed, including three soldiers.
The Kenyan Red Cross says 51 people are still missing, and the BBC’s Gabriel Gatehouse says mortuaries in the capital are expecting to receive more bodies.
There are reports that part of the roof at the Westgate shopping centre has collapsed, following a fierce blaze on Monday.
The Somali Islamist al-Shabab movement said it had carried out the attack in retaliation for Kenyan military operations in Somalia.
Somali Prime Minister Abdi Farah Shirdon has said that those behind the attacks “must be held accountable.”
Meanwhile, Kenya’s foreign minister said “two or three” Americans and a British woman were among the attackers.
In an interview with the US TV programme PBS Newshour, Amina Mohamed said the Americans were 18 or 19 years old, of Somali or Arab origin, and lived “in Minnesota and one other place”.
She said the Briton was a woman who had “done this many times before”.
Ms Mohamed appeared to contradict earlier comments from Kenya’s interior minister, who suggested that all the attackers were men – though some may have been dressed as women.
Ms Mohamed’s remarks have fuelled media speculation about the possible involvement of Samantha Lewthwaite, the widow of one of the men who carried out attacks on London’s transport system on 7 July 2005.
British officials said they would not be drawn on the identity of the attackers while investigations continued.
Mosque leaders in Minnesota have condemned the violence, saying the attacks had nothing to do with Islam.
The Kenyan security forces are continuing to comb the shopping centre to flush out the last of the militants.
The interior ministry said security officials were in “mop-up operations” and that “we’re very near the end”, while the police said they were defusing explosives planted in the area.
The BBC understands from official sources that six of the attackers have been killed – three on Monday and three since midnight.
The government said there were 10 to 15 militants involved in Saturday’s attack, and it is not yet clear how the rest are accounted for.
Al-Shabab has made counter-claims on social media that it is still in contact with fighters in the shopping mall, and they still have hostages.
The government has said it has no indication that there are any captives.
Officials said earlier that 10 people had been arrested in connection with the attack.
Militants stormed the Westgate centre on Saturday, throwing grenades and firing on shoppers and staff.
At least 18 foreigners are among the dead, including six Britons, as well as citizens from France, Canada, the Netherlands, Australia, Peru, India, Ghana, South Africa and China.
The interior ministry said on Tuesday that three soldiers had succumbed to their injuries, and eight were still being treated. “We’ve lost three heroes,” the ministry said on its Twitter account.
Nearly 200 people were wounded.
President Barack Obama called the attack a “terrible outrage” and said the US was providing all the co-operation it could to Kenya.
Thousands of Kenyans have been responding to appeals for blood donations.
Al-Shabab, which is linked to al-Qaeda, 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 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 remains in control of smaller towns and large swathes of the countryside.
UN special representative for Somalia Nicholas Kay called on Tuesday for a fresh surge in African troops to Somalia to counter an estimated 5,000 al-Shabab fighters, Reuters news agency reported.
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