South Sudan's Information Minister Barnaba Marial Benjamin told reporters in Juba the aerial bombardment of the facility - in the Heglig region - had caused serious damage.
"They are bombing the central processing facility and the tanks to rubble as we speak," he said.
Sudan's state minister for information denied the charge, however, telling Al Jazeera television that Sudan "did not and will not" destroy the oil facilities.
Fighting between the two neighbors has already halted production at the facility, depriving Sudan of about half of its 115,000-barrel-a-day oil output.
Both sides regularly make conflicting claims and limited access to the remote region makes it difficult to independently verify their statements.
But the seriousness of the allegations underscored how close the two former civil war foes are to the brink of a full-blown war as the worst fighting since Juba declared its independence in July continued apace.
On Tuesday, South Sudan seized control of the Heglig oilfield, drawing an angry response from Khartoum which vowed to recapture the region.
On Sunday, South Sudan's military (SPLA) spokesman Philip Aguer said by phone that Sudanese aircraft were continuing efforts to dislodge southern forces from the area, subjecting them to an aerial bombardment.
"There has been intensive air bombardment against our position in Heglig since the morning," he said, adding there had been no ground fighting there on Sunday.
Sudan's army had also shelled a western part of South Sudan's Upper Nile state, in an apparent attempt to open a new front, he said.
A spokesman for Sudan's armed forces did not immediately answer calls to his mobile phone for comment. The Sudanese army said it had entered the Heglig region on Saturday, but South Sudan denied that.
Worsening violence in recent weeks has all but killed hopes that the two countries will reach a swift agreement on disputed issues such as the demarcation of their 1,800-km (1,200-mile) border, the division of national debt and the status of citizens in each other's territory.
Sudan has already pulled out of talks over those and other issues. It says it will not return to the negotiating table until the South pulls out of Heglig.
"Sudan reiterated its stated and fixed position that it will not negotiate with South Sudan unless it withdraws its forces from the Heglig region," state news agency SUNA reported on Sunday, citing President Omar Hassan al-Bashir.
EGYPT STEPS IN
The two sides fought one of Africa's longest and deadliest civil wars, which ended in 2005 with a peace deal that paved the way for the South's independence
Egypt, which borders Sudan to the north, is mounting a diplomatic offensive to try to defuse the current tensions.
Egypt's Foreign Minister Mohamed Kamel Amr arrived at Khartoum airport on Sunday for talks.
"We are trying to preserve blood and resources and find a peaceful solution, because this issue does not just concern the two countries, but all their neighbouring countries," he told reporters in Khartoum, adding he will travel to Juba on Monday.
Global powers have widely condemned South Sudan's seizure of Heglig, urging the two sides to stop fighting and return to talks. South Sudan says Heglig, which many southerners refer to as Panthou, is its rightful territory, an assertion Khartoum hotly contests.
The border clashes come as Sudan is battling armed insurgencies in its western Darfur region as well as in its border states of South Kordofan and Blue Nile.
Khartoum accuses South Sudan of supporting the rebels in those areas. Juba denies that.
Both nations are also under increasing economic pressure from a loss of oil revenues.
Landlocked South Sudan shut down its own output - about 350,000 barrels a day - in January after failing to agree how much it should pay to export crude via pipelines and other infrastructure in Sudan.
Inflation has risen sharply since, while both currencies have depreciated on the black market.
Sabir Hassan, one of Sudan's top negotiators, said talks on economic issues were impossible amid the current tensions.
"This is a climate of war," he told Reuters. "The best thing is for the south to stop this policy, and for the two sides to sit down and try to negotiate ... and try to live in peace."
"War is imminent between the two countries if that policy continues in the south," Hassan added.
(Additional reporting by Yasmine Saleh and Khalid Abdelaziz)
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