NAIROBI, KENYA — Seeking to end a siege that has left scores dead and shaken the nation, the Kenyan government said Sunday night that it was pressing an assault against al-Shabab militant attackers who had been holed up in a Nairobi shopping mall for more than a day.

Late Sunday, the Kenyan military announced that it had retaken “most” of the Westgate mall — the attackers had been confined to the third floor since their initial assault on Saturday — and freed more hostages, though details could not be confirmed. Helicopters circled the mall building through the night, and occasional explosions and bursts of gunfire were heard above a rainstorm in the area.

“This will end tonight — our forces will prevail,” the police command center said in a Twitter post. “Kenyans are standing firm against aggression, and we will win.”

Al-Shabab, a militant group mostly based in neighboring Somalia, answered with messages of their own, including warning that “Kenyan forces who’ve just attempted a roof landing must know that they are jeopardizing the lives of all the hostages at Westgate.”

Later, officials said that at least four members of the security forces had been wounded. But there were no other details about additional casualties on either side.

The attack on the mall deeply distressed Kenya, a nation that has grown in stature as a force against terrorism in East Africa. As the toll mounted — at least 68 were reported dead by late Sunday, with several people still unaccounted for —the potential for even greater loss of life seemed tangible.

Addressing the nation, President Uhuru Kenyatta sounded a note of solidarity in loss, revealing that his nephew and the man’s fiancée were among the dead. “These are young, lovely people I personally knew and loved,” Kenyatta said. “Many of us have lost loved ones. Let us mourn them all as one nation and keep them always in remembrance and prayer.”

He said security forces had rescued more than 1,000 people from the mall since the violence began Saturday, calling the forces’ performance “remarkable and encouraging,” even as he asked for patience from the public as the standoff continued.

The assault on Westgate was the deadliest terrorist attack in Kenya since the 1998 al-Qaida truck bombing of the U.S. Embassy in Nairobi that killed more than 200 people.

President Barack Obama called Kenyatta on Sunday to reaffirm the “strong and historic partnership between the United States and Kenya.” That relationship has been strained by the election in March of Kenyatta, who is being prosecuted at the International Criminal Court on charges of financing death squads during an outbreak of political violence in 2007. Obama skipped visiting Kenya, his father’s birthplace, on his trip to Africa in late June.

Al-Shabab has said that they staged the mall attack as retribution for the Kenyan military presence in Somalia, where Kenyan troops have driven al-Shabab fighters out of much of the territory they once controlled. A confidential U.N. security report described the assault on the mall as two-pronged, with groups of gunmen attacking on different floors simultaneously.

Joseph Ole Lenku, the Cabinet secretary for the interior, said 10 to 15 attackers were inside the mall. Kenyatta said he could neither confirm nor deny reports that one or more of the attackers were women. Al-Shabab claimed in a Twitter post that three of the attackers were Americans; a senior law enforcement official in the United States said that the FBI had yet to establish whether that claim was true, and that it would be difficult to do so until all the attackers were captured or killed.

The number of bystanders remaining in the building was not as clear, though the Kenya Red Cross, citing the police as its source, said Sunday that 49 people were unaccounted for, raising the prospect of a significantly higher death toll before the crisis ends.

The American official said that the FBI had offered assistance to the Kenyan authorities and that FBI agents were at the Kenyan command post at the scene. There were reports that Israelis were supporting the Kenyan authorities as well; a spokeswoman for the Israeli Foreign Ministry said she could not comment. The secretary-general of Interpol, Ronald K. Noble, said in a statement that his agency had offered to send a response team, including forensic and counterterrorism experts.

Kenyatta said that he had received “numerous offers of assistance from friendly countries; for the time being, however, this remains an operation of the Kenyan security agencies.”

Among those killed in the mall were three Britons, the British Foreign Office confirmed Sunday. Five Americans were among the wounded, but none were known to have been killed. News agencies reported that other foreigners were also among the dead.