South Sudan won independence from Khartoum in 2011, making it the world's newest country.
But critics say that Kiir's militarised and authoritarian Government has failed to establish a functioning state, despite possessing nearly 75 per cent of all the oil found within the previous borders of Sudan.
Juba's international airport opened yesterday for the first time since the violence began. Scheduled flights and chartered aircraft were full of people trying to flee.
Meanwhile, a convoy guarded by soldiers, including at least 14 buses filled with fleeing expatriates, crossed the border into neighbouring Uganda.
A Western security adviser in Juba said the city was "calmer today than it has been for the last couple of days", adding that traffic was moving freely and "some shops and stalls are open".
He said: "But I get the sense that people are rushing out to stock up, then they're going back inside."
Kiir said he was willing to "sit down" for talks with his former Vice-President. But Machar denied any attempt to mount a coup and said that he was being targeted for opposing the President. "There was no attempted coup. Someone wanted to frame me. I had to flee, they are hunting me down," he told the BBC.
Machar and Kiir are the political leaders of South Sudan's two largest ethnic groups, the Nuer and the Dinka respectively. Commentators have warned for months that the growing antipathy between the two men could eventually spiral into tribal conflict between their supporters.
"Salva must recognise that the charge of his being 'dictatorial' has taken deep hold, and he must do what is necessary to shed the label as much as possible," said Eric Reeves, a South Sudan expert. "There are a number of urgent steps he must take to secure the confidence of the international community, and to put the Nuer community at ease in the Juba area and elsewhere.