At least one man was shot dead, witnesses said.
President Mwai Kibaki made the statement to Jendayi Frazer, the leading U.S. diplomat for Africa, according to the director of the presidential news service, Isaiya Kabira.
Kabira offered no details and would not say whether it was a formal offer to Raila Odinga, the opposition leader who accuses Kibaki of stealing the Dec. 27 vote described by international observers as deeply flawed.
Hour later, Odinga's spokesman told CBS News reporter Katherine Arms that they had dropped all preconditions to international mediation, and they wanted the head of the African Union to mediate.
This is now Kibaki's move, Arms said, because he has to ask him to come.
Britain, the former colonial power in Kenya, appealed Friday to leaders in the East African country to consider sharing power.
Frazer, who met with Odinga earlier Saturday, would meet with the opposition leader again, Kabira said.
Odinga told a news conference he had not received any formal offer from the government, but added, "Let them put that on the table when we are negotiating."
He declined to give a response, but his spokesman, Salim Lone, told The Associated Press that Odinga would rather not share power.
"Raila has said a number of times that he is not happy with (the idea of) a government of national unity, he has said he would rather remain in the opposition," Lone said.
Odinga spoke at the news conference before he met for a second time with Frazer on Saturday. Aides would not say whether Frazer gave Odinga a message from Kibaki. Lone would only say that Odinga repeated his demand for a new election.
In parliamentary balloting, Odinga's party won 95 of 122 legislative seats and half of Kibaki's Cabinet lost their seats, making it almost impossible for Kibaki to govern without opposition cooperation.
There was no immediate statement from Frazer on her 90-minute meeting with Kibaki or her talks with Odinga.
Kabira read a government statement that quoted Frazer as saying that "by extending an olive branch to the opposition, President Kibaki had shown his commitment to ending the political impasse."
The statement said Kibaki was ready to work with all involved parties.
"The president said he was ready to form a government of national Unity that would not only unite Kenyans but would also help in the healing and reconciliation process," the statement said.
U.S. State Department spokesman Sean McCormack said Frazer's mission is designed to complement other international efforts, including one by South African Nobel peace laureate Desmond Tutu, toward a peaceful solution.
French Foreign Minister Bernard Kouchner has said that the elections "were totally rigged," but did not provide evidence.
Attorney General Amos Wako has called for an independent investigation of the vote counting. The call from Wako, who is considered close to Kibaki, was a surprise and could reflect the seriousness of the rigging allegations.
But Odinga's spokesman, Lone, rejected the suggestion, saying his party had "no faith in any government institution."
The state-funded Kenya National Human Rights Commission and 22 other civil society organizations called Saturday for the entire electoral commission to resign for participating in "a forgery" of an election.
The United Nations said 250,000 have been displaced because of violent protests and clashes since the vote.
The death toll looms at about 350 people, but this is sure to rise when people are found behind closed doors in the slums and out in rural areas where the press have not been able to reach.
The turbulence has taken an ugly ethnic twist, with other tribes pitted against the president's Kikuyu people, and brought chaos to a country once considered an island of stability in violence-plagued East Africa.
Several shacks were set ablaze Saturday in Nairobi's sprawling Mathare slum, where residents attacked each other with machetes. One man said people from Odinga's Luo tribe were fighting a gang from Kibaki's Kikuyu tribe. Police opened fire and one man was shot in the head and killed, according to an AP Television News cameraman.
The police were quickly surrounded by an angry crowd and fled with three wounded people, including a man with half his leg hacked away.
Trouble has spread from Nairobi, the capital, to the western highlands and to the coast.
In the coastal tourist city of Mombasa on Saturday, police fired tear gas in a bid to disperse protesters for a second day running.
"Kibaki must go!" the scores of demonstrators shouted.
Thousands in the capital's slums, meanwhile, have lined up for food after days of riots left them cut off and shortages have led to major price increases.
In Eldoret and areas to the west, the Red Cross is distributing food from the World Food Program to an estimated 100,000 people.
The U.N. World Food Program said it was scrambling to bring food to 100,000 displaced people in the Rift Valley. The agency said trucks were slowed because of insecurity.
WFP Spokesman Marcus Prior said, "These people need so much assistance. They fled with absolutely nothing and many have lost everything."
In Nairobi, appeals have been made to help the homeless. People from all areas here have flocked to give money and goods to those who so badly need it.
Arms reports that there are many more cars and trucks on the roads today. Also, loads of people on foot are carrying belongings and heading for areas in which they would feel safer.
Associated Press writers Malkhadir M. Muhumed, Elizabeth A. Kennedy, Tom Maliti, Katharine Houreld and Todd Pitman contributed to this report.