Omar al-Bashir, Sudan's president, has pledged not to negotiate with South Sudan amid reports of fresh air attacks on his country's southern neighbour.
The renewed tensions come as South Sudan's 10-day occupation of the oil town of Heglig has left parts of the town blood-soaked and in ruins.
"We will not negotiate with the South's government, because they don't understand anything but the language of the gun and ammunition," Bashir told Sudanese troops at a barracks near an oilfield along the two neighbours' contested border on Monday.
"Our talks with them were with guns and bullets."
General Kamal Abdul Maarouf, a Sudanese army commander who led the battles in Heglig, said the army had killed 1,200 South Sudanese troops in fighting in the area, an account South Sudan denied.
An AFP news agency correspondent who accompanied Maarouf said he saw piles of corpses bearing South Sudanese military uniforms scattered beneath trees in the border region.
South Sudan's army said 19 of its soldiers were killed and that 240 Sudanese troops lost their lives.
'So many bodies'
Early in the occupation, one South Sudanese soldier in Bentiu, capital of the South's Unity state, said "there are so many bodies at the front line, so many dead", that it is impossible to bury them or bring them back.
Despite the end of the occupation, Major General Mac Paul, the deputy director of military intelligence for South Sudan, said on Monday that two MiG 29 fighter planes dropped three bombs, two of which landed near a bridge that connected Bentiu and Rubkona.
"This is a serious escalation and violation of the territory of South Sudan. It's a clear provocation," he said.
|In-depth coverage of North-South strife over border
Paul said ground troops from Sudan launched three waves of attacks about 10km on its side of the border.
"We are building up troops because we think that the Sudanese army is also building up," he said.
A Reuters journalist saw aircraft dropping two bombs near a bridge linking two areas of Bentiu, although it was not possible to verify the planes' affiliation.
Sudan denied carrying out the air raid, which drew quick condemnation from the US and France.
The military "denies any sort of direct bombing inside the border of South Sudan", an official from Sudan's foreign ministry said.
The governor of Unity state, Taban Deng, said the Sudanese bombs fell on a key bridge and a market, killing at least two children. The Reuters journalist reported seeing the body of one child.
The bombs prompted heavy bursts of gunfire from Southern soldiers hoping to shoot down enemy fighter jets.
In the market, stalls were on fire and large plumes of grey smoke rose high into the air, as screaming civilians ran in panic.
Deng said that the violence was a result of his country's pullout from Heglig.
"We have been pressured by the international community to pull out of Heglig and this is the consequence, we have brought the war to home," he said.
"They have been given orders to wipe us out, they have called us insects," Deng said, referring to Bashir's earlier speech.
Al Jazeera's Zeina Khodr reports from Khartoum
Bashir and his defence minister, Abdelrahim Mohammed Hussein - both wanted by the International Criminal Court for alleged war crimes in Sudan's Darfur region - declared on Friday that their army had forced Southern soldiers out of Heglig.
Salva Kiir, South Sudan's president, had already announced that his forces would leave under "an orderly withdrawal". His army said the pullout was completed on Sunday.
A US monitoring group said on Sunday that satellite imagery appeared to show the fighting around Heglig had caused major damage to oil-pipeline infrastructure.
The Satellite Sentinel Project said the images showed severe damage, and in such a critical part of the oil infrastructure that it would probably stop oil flow in the area.
From the main road in Heglig, destroyed oil-company vehicles could be seen, and the AFP correspondent found the area's main oil-processing facility heavily damaged.
A storage tank appeared to have been destroyed by fire; eight generators which provided power to the facility were damaged; and some oil was leaking onto the ground at the plant operated by the Greater Nile Petroleum Operating Company (GNPOC).