Sudan and South Sudan both want to avoid "all-out war", a US envoy said on Thursday, despite the Sudanese president's threat to topple the Southern regime which seized its main oil field.
Princeton Lyman, US special envoy on Sudan and South Sudan, said the two sides wanted a way out and he was pressing the South to withdraw from the Heglig oil field.
Diplomatic efforts to end the deadly standoff and prevent a wider conflict have intensified.
"I can say with confidence that virtually everyone I've talked to has said, look, we don't want to go to all-out war with the other," Lyman told reporters by telephone from Khartoum, after talks in the two countries.
Lyman acknowledged the neighbours were in a "very, very serious crisis" but said both sides were mindful of international pressure.
Keeping up war rhetoric that has sparked concern from Washington, a beret-wearing Bashir told a rally of freshly-trained paramilitary troops, some riding camels, that Sudan will teach the Southern government "a lesson by force."
A day earlier he called the Juba regime an "insect" that Sudan aims to eliminate.
On the battlefield, the Southern army repulsed ground attacks in a widening conflict zone along the border, said the South's military spokesman Philip Aguer.
Ground assaults and air strikes were reported in the Heglig region, as well as three Southern border states, he said, adding the Southern army was "still in its positions".
Sudan has vowed to reclaim Heglig "by hook or crook", but its military has released virtually no information about the situation on the ground.
Rebels from Sudan's Darfur region said they seized two Sudanese military positions 40 kilometres (25 miles) north of Heglig. But the insurgents from the Justice and Equality Movement deny fighting alongside the South Sudanese.
AFP reporters who have visited the Heglig battle zone along the disputed border said dead bodies and destroyed tanks were strewn about.
East Africa's main diplomatic body, the Inter-Governmental Authority on Development (IGAD), expressed "grave concern about the escalating conflict".
The IGAD, which mediated peace talks that ended Sudan's 1983-2005 civil war in which about two million people died, said it was ready to provide "all possible assistance to maintain peace and stability."
On Tuesday, the UN Security Council discussed possible sanctions against both Sudan and South Sudan in a bid to halt a wider war.
Clashes escalated last week with waves of air strikes hitting the South, and Juba's seizure of the Heglig oil hub on April 10.
South Sudan's Information Minister Barnaba Marial Benjamin said in Juba that his country was not at war with Khartoum and was interested in peaceful relations.
"The Republic of South Sudan considers Sudan as a neighbour and friendly nation, not an enemy," Benjamin told reporters.
China, which has a vested interest in the oil industries of both nations, also reiterated its concern over the escalating conflict and called for a resumption of dialogue.
Khartoum's parliament has declared a "mobilisation and alert" of the population and this week voted unanimously to brand the South Sudanese government an enemy.
Benjamin, the Southern minister, said the move amounted to a "declaration of war". Sudan was using Heglig as a base to launch attacks on his country, he alleged.
The UN, the US and the European Union have criticised the South's occupation of Heglig but have equally denounced Sudan's air strikes against the South.
Lyman said he told President Salva Kiir and other officials in the South's capital, Juba, that they should "note the unanimous reaction of the international community."
The fighting is the worst since South Sudan became independent last July, with strong US support.
Although South Sudan disputes it, Heglig has been internationally regarded as part of Sudan.
Since the invasion, production at Heglig has been shut and facilities there are leaking. The field accounted for about half of Sudan's oil, and its loss has worsened an economic crisis.
The International Monetary Fund on Wednesday forecast that Sudan's economy will shrink 7.3 percent this year, while consumer prices are expected to rise by an average of 23.2 percent.