A South Sudan official, meanwhile, said the fighting was "spreading all over".
The two sides fought a civil war that lasted decades, and any increase in sporadic border clashes raises the risk of a return to all-out war.
The Sudanese Colonel Sawarmy Khaled told Sudan's state radio that the South's army had attacked the border town of Heglig twice in the past 24 hours. South Sudan officials would not confirm whether their troops were in control of the oilfields.
Hostilities between Sudan and South Sudan have grown in recent months, even as the south has said it is trying to avoid a return to war. The two sides never reached a deal to share the oil resources in the region or the exact location of the border, which has added to the tensions.
South Sudan's army - the SPLA - said it moved into Heglig on Tuesday after repelling an attack launched by Sudanese Armed Forces against an SPLA position near the border town of Teshwin.
An SPLA spokesman said Sudanese MiG-29 fighter jets bombed the area on Monday and Tuesday. He said several SPLA soldiers were injured in the attack but would not say how many.