The Council is now considering sending a multi-faith team of Christian and Muslim leaders to the capital Khartoum.
The clashes between southern Sudanese and northern Arabs followed Saturday's death of southern leader John Garang.
The US has expressed deep concerns over the situation. Its two envoys are due to meet President Omar al-Bashir.
Most of the violence has occurred in the capital, Khartoum, where at least 84 people have been killed.
A curfew was in place in Khartoum overnight - for a third night running.
At least 18 people are reported to have been killed in the southern town of Juba, where Garang had his headquarters.
Northerners - many of whom have lived in the south for decades - are said to be packing up and leaving the city.
Leaders of both sides earlier urged calm, agreeing to set up a joint inquiry into what caused the crash of Garang's helicopter.
But correspondents say there is a real danger that a much-vaunted peace deal ending 21 years of civil war could be shattered by the sudden violence.
Mr Kiir needs all the help he can get, says the BBC's Peter Greste in New Site, southern Sudan. He has not even been sworn in yet as the new leader of the SPLM movement and holds no formal power.
That is why he asked a Council delegation visiting New Site to pay their respects to the late John Garang to send envoys to Khartoum and appeal for calm, our correspondent says.
He adds that the southern administration is also hoping that a joint team of investigators will confirm the helicopter crash was indeed an accident and not an assassination, as some of Garang's followers suspect.
Garang signed a deal to end the civil war in January and was sworn in as vice-president of Sudan three weeks ago.
He is due to be buried in Juba on Saturday, after his body is taken through other key towns in southern Sudan to allow supporters to pay final respects.
Local media say President Bashir is expected to attend the funeral.
- Addis Tribune
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