Ban Ki-moon: "I spoke to President Salva Kiir yesterday morning urging him to do everything possible he can to end the violence"
Fighting is spreading in parts of South Sudan after a reported coup attempt in the capital, Juba, at the weekend.
Violence has erupted in the flashpoint town of Bor, capital of eastern Jonglei state, and in Torit, capital of Eastern Equatoria.
President Salva Kiir has accused ex-vice-president Riek Machar of staging the coup - a claim he denies.
The UN called for political dialogue to end a crisis that has left hundreds dead and sparked fears of a civil war.
On Wednesday, Mr Kiir said he was willing to enter into talks with Mr Machar but that he did not know what the result would be.
'Panic'
UN Secretary General Ban Ki-moon said he was "deeply concerned" about the situation in the country.
He said: "This is a political crisis and urgently needs to be dealt with through political dialogue. There is a risk of this violence spreading to other states... [and] we have already seen some signs of this."
Mr Ban confirmed that some 20,000 people had taken refuge in the UN mission in Juba.
The mayor of Bor, Nicholas Nhial Majak, told the BBC's Focus on Africa programme that violence had spread there from Juba, where it began on Sunday. Bor is about 150km (90 miles) north of the capital.
The mayor said troops loyal to breakaway army commander Peter Gadet had raided the town and there was "a lot of fighting".
"There is a lot of panic in town - and we really want the authorities to send more troops to quickly restore hope for the people," he said.
The South Sudan Red Cross Society said at least 19 people were killed in fighting there.
Joe Contreras, spokesman for the UN Mission in South Sudan (UNMISS), said: "Hundreds of civilians have been streaming into our camp on the outskirts of the town, they're now over the 1,000 mark, and Bor is very tense."
One witness in the town saw a police guard shot dead and said that rebel troops had raided the police and prison headquarters, and blocked the road to Juba.
South Sudan military spokesman Col Philip Aguer admitted that "we don't know who is fighting who" in Jonglei state.
The UN has expressed fears of a civil war between the two main ethnic bodies, the Dinka - Mr Kiir's group - and the Nuer, from which Mr Machar hails.
Gunfire continued in Juba in the early hours of Wednesday but the city was later reported to be quieter.
Businessman Muhammad Waqas ur-Rehman told the BBC the city was tense and that he and many others were trying to leave.
Presidential spokesman Ateny Wek Ateny told the BBC that government forces were in full control of Juba and all other cities, and that although there were "pockets of resistance", there were no major engagements taking place.
Mr Ateny played down the prospect of civil war and said that while it was likely that civilians had been caught in the fighting, no-one was "targeting ethnic groups".
BBC world affairs correspondent Mike Wooldridge says it remains to be seen what impact such government assurances will have in what has been a tense and volatile situation.
The UK's Foreign Office said it was pulling some of its embassy staff out of Juba "due to the current instability".
Salva Kiir told a press conference he was ready to talk to Riek Machar
The UN says 20,000 have sought shelter in Juba
People gather at Juba airport to catch flights out
It said the embassy remained open but that it now advised "against all travel to Jonglei state and Juba".
The US has suspended embassy operations and on Wednesday flew out 120 evacuees, including 28 foreign diplomats, on two C-130 aircraft.
South Sudan Information Minister Micheal Makuei Lueth confirmed to the Associated Press that at least 500 people, mostly soldiers, had been killed "in the bushes" around Juba.
He said up to 700 more had been wounded.
Earlier Mr Machar denied allegations that he had tried to stage a coup.
He told the BBC World Service's Newsday programme: "Salva wanted to frame me. I had to flee. They are hunting me down."
The whereabouts of Mr Machar are unclear. He told the BBC he was still in South Sudan and was "not going to leave the country".
He blamed Sunday's fighting on a conflict between members of the presidential guard.
He added that government troops had used the incident to arrest some of his supporters.
Mr Kiir said a group of soldiers supporting Mr Machar had tried to take power by force on Sunday night but were defeated.
South Sudan has struggled to achieve a stable government since becoming independent from Sudan in 2011.
The oil-rich country remains ethnically and politically divided, with many armed groups active.
After a peace deal was signed in 2005, Mr Machar was appointed vice-president of the South Sudan regional government.
He retained the position after independence in 2011 but was dropped in July when the whole cabinet was sacked.
Both Sudan and the South are reliant on oil revenue, which accounts for 98% of South Sudan's budget. They have fiercely disagreed over how to divide the oil wealth of the former united state - at one time production was shutdown for more than a year. Some 75% of the oil lies in the South but all the pipelines run north.
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