Rebels are said to be in control of some of South Sudan's oilfields, raising questions of how long the country's oil will flow and whether Sudan could enter the conflict.
South Sudan President Salva Kiir has implored his country to turn away from ethnic violence and has met foreign ministers from neighbouring states, including Kenya and Ethiopia, who flew into the capital, Juba.
They are trying to calm tensions after a week of ethnic strife that is estimated to have killed hundreds of people.
Kiir said via the South Sudanese government's Twitter feed: "Those who may want to take the law into their hands, the long arm of the government will get them."
On Saturday, Australian Foreign Minister Julie Bishop urged Australians to leave the country.
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"I am aware there are a number of Australians who remain in South Sudan. However, I have not received any reports from local authorities of any Australians being injured as a result of the violence," Ms Bishop said.
The UN Security Council expressed "grave alarm" at the rapidly deteriorating security crisis, condemned targeted ethnic violence and demanded an end to the fighting.
The UN's most powerful body urged the president and ousted vice president Riek Machar "to demonstrate leadership in bringing a swift and peaceful resolution to this crisis".
France's UN ambassador, Gerard Araud, the council president, told reporters that Kiir and the widow of South Sudan's rebel hero John Garang, who led the country's fight for independence, had agreed to enter an unconditional dialogue.
There was no word yet from Machar, who is believed to be in hiding.
Kiir, an ethnic Dinka, earlier this week said an attempted coup had triggered the violence, and the blame was placed on Machar, an ethnic Nuer.
But officials have since said a fight between Dinka and Nuer members of the presidential guard triggered the fighting on Sunday night.
Fighting continued to spread in Jonglei and Unity state, an oil area, as armed groups opposed to the nation's military emerged, said a South Sudan expert communicating with combatants and UN officials in strife-torn regions outside the capital.
"We've seen the emergence of different armed groups under different commands, and we've seen the former vice president say he's not interested in talks that don't end in Salva Kiir stepping down," said Casie Copeland, South Sudan analyst for the International Crisis Group.
