NAIROBI, Kenya— Authorities said they were going floor by floor through a darkened, smoldering mall early Tuesday as they searched for remnants of a team of attackers that had seized the building, and that they believed all those trapped or held hostage had been freed.

U.S. officials said they could not confirm reports that Americans were among the estimated 10 to 15 attackers. The al-Shabab militant group, which claimed responsibility for the assault, is linked to al-Qaida. It is known to have recruited Americans, Europeans and other foreigners, including Kenyans.

Officials, who reported Sunday that 68 people had been killed, lowered the toll to 62 on Monday, saying some of the dead had been counted twice. Among those killed were six British citizens; two French; two Canadians; and a well-known Ghanaian poet, Kofi Awoonor.

Kenyan President Uhuru Kenyatta said a nephew and his nephew’s fiancee had been slain, but that more than 1,000 people escaped the upscale mall after the attack began midday Saturday.

A major assault by security forces began with gunfire ringing out at dawn Monday. The attack reached a crescendo in the early afternoon, with a volley of explosions and heavy arms fire. Black smoke poured out of the building, as a fire raged for many hours.

Kenyan armed forces chief Gen. Julius Karangi said the fire was started by the assailants as a diversion in a bid to flee the building, but provided no more detail.

Explosions and gunfire continued on and off throughout the day. Defense officials said the attackers were in one location in the mall, with no way to escape.

Earlier Monday, Interior Minister Joseph Ole Lenku said at a news conference that efforts to release the hostages had been “very, very successful” and those being held were “very few.”

But it wasn’t clear whether people had simply escaped the mall after hiding or whether they had been freed by security forces. There was also speculation that the attackers may have been holed up in a supermarket within the mall, without hostages.

As of late Monday, it appeared no one had come forward to say they had been held by the militants and freed by security forces. Several people described Sunday how they had hidden in the mall and later escaped.

Al-Shabab said it attacked Kenya’s most prestigious shopping center to punish the country for sending soldiers into Somalia. The group had been warning for two years that it would mount a major terrorist assault in Nairobi, but to date its attacks have largely been small-scale shootings or grenade attacks.

The group had controlled most of Somalia, which has been without a functioning government for much of the last 20 years. But it has lost most of its strongholds since Kenya invaded. An African Union military force is also working with Kenya’s weak national government.

Ben Rhodes, President Barack Obama’s deputy national security adviser, said the U.S. had been concerned for some time about efforts by al-Shabab to recruit Americans to fight in Somalia. Americans are known to have joined the group.

“This is an issue that has been tracked very closely by the U.S. government, and it’s one that we’ll be looking into in the days ahead,” he said.

Rhodes said the U.S. was determined to continue to pressure al-Shabab. He said Obama had called Kenyatta to offer assistance with the investigation of the attack and efforts to confront al-Shabab.

“But the clear message that we need to send in response is the resolve of Kenya, the international community and the United States will not be shaken at all, and, in fact, our determination will only be increased to keep the pressure on al-Shabab and to make sure that they cannot have a safe haven in Somalia to launch these types of attacks,” Rhodes said.

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