The stand-off follows clashes along the ill-defined border that have brought the former civil war foes closer to returning to full-blown conflict than at any time since South Sudan declared independence last year.
South Sudan seized the Heglig oil field on Tuesday, drawing international condemnation including rare criticism from the United States. Sudan has called the move a "blatant assault" on its sovereignty and demanded an immediate withdrawal.
Heglig, which the south claims as its own, is vital to the north's economy because the field there accounts for about half of the country's remaining 115,000 barrel-a-day oil output.
Ahmed Haroun, governor of Sudan's South Kordofan border state, said crude production had stopped in Heglig, but the army was "dealing with the situation".
"We hope we can finish that operation in hours ," he told reporters in the town of Talodi in South Kordofan, a state on the border with South Sudan where Khartoum is also battling insurgents who it alleges are aligned with Juba.
With the European Union voicing "deep concern" at the escalating conflict, the north said it was mobilising its army to retake the Heglig field, which the South said it had seized to end attacks from Khartoum.
South Sudan's President Salva Kiir said the conflict could lead to war.
"I urge you to sensitise our people to the probability of a full-scale military aggression by the Republic of Sudan against the Republic of South Sudan," he told parliament, according to a transcript of the speech.
"The intention of Khartoum is to take over the Unity oil fields," he said, adding Heglig was southern territory.
Kiir said he was still ready to meet Sudan's President Omar al-Bashir in Juba or elsewhere after Sudan called off a summit that had been scheduled for April 3.
South Sudan's Information Minister Barnaba Marial Benjamin also quoted Kiir as saying South Sudan's army (SPLA) would not withdraw without guarantees from the international community.
"There must be a mechanism so they (Khartoum) don't launch another attack," Benjamin told Reuters.
He said that Sudan's air force had dropped six bombs near the oil town of Bentiu in Unity State south of the border on Thursday, killing one soldier. Sudan denied the charge.
The violence has dampened hopes that Khartoum and Juba would soon reach a deal over oil payments and other disputed issues. Sudan pulled out of the negotiations on Wednesday after South Sudan seized Heglig.
UN Secretary-General Ban Ki-moon urged both governments to put an end to the fighting and resume talks.
"In this moment of inflamed tensions, it is imperative that both sides exercise the maximum restraint, in the interests of their peoples and their common future," he said in Geneva.
But there were few signs of an immediate halt to the violence. Clashes also broke out in Blue Nile, another of Sudan's border states, according to rebels operating there.
The insurgents - members of the southern army during Sudan's civil war who say they have since severed links with Juba - claimed they had killed 13 government troops and destroyed military vehicles in a six-hour battle near the state capital.
Sudan's armed forces spokesman, Al-Sawarmi Khalid Saad, dismissed the battle as a "small clash" between two patrols and said one Sudanese soldier had died while 11 rebels were killed. Neither claim was possible to verify independently.
Officials in the north have accused South Sudan of "using mercenary forces and rebel groups" in its attack on Heglig and decried what they said was a plot, sponsored by Juba's foreign allies, to overthrow the government in Khartoum.
The South seceded from Khartoum's rule last year, but the two sides have not agreed on issues including division of national debt, the status of citizens in each other's territory and the exact position of the border.
Landlocked South Sudan shut down its roughly 350,000 barrel-per-day output in January in a dispute over how much it should pay to export crude via pipelines and facilities in Sudan.
Sudan's Bashir said foreign forces were fomenting conflict.
"War is not in the interests of either people and they have chosen the path of war in the implementation of foreign agendas," Bashir told reporters in Khartoum.
Motorists fearing disruption of fuel supplies formed lines at petrol stations on Wednesday to stock up after news of the Heglig attack reached Khartoum. The government said there was no threat to supplies.
"The national economy is stronger than the aggressors think and Sudan has enough stockpiles of basic goods," Sudan Council of Ministers spokesman Omer Mohamed Saleh said in comments relayed by online newspaper the Sudan Tribune.
South Kordofan governor Haroun said all Sudan's other fields were secure. "They are working as usual," he said, adding that more details about Heglig's situation would become clear after fighting had stopped and technicians entered the area.
EU foreign policy chief Catherine Ashton said South Sudan's move on Heglig was "completely unacceptable" and chided the north for aerial bombardment of southern territory.
"Both parties must also stop supporting armed groups in the territory of the other state," Ashton said in a statement.