Sudan has threatened to stop all its security and economic agreements with South Sudan if it continues its alleged support for rebel groups.
Khartoum also warned it would cut off its southern neighbour from vital oil pipelines.
"We plan to close the oil pipelines within 60 days," Information Minister Ahmed Belal Osman said on Sunday.
"But if South Sudan is serious ... and stops backing rebels, if we get international guarantees for that, then our door is open and we can reverse the stoppage."
His comments followed an order on Saturday from President Omar al-Bashir to shut the pipeline carrying South Sudanese crude for export.
A stoppage would cut off the crude and transit fees that make up both countries' main source of foreign income.
Al Jazeera's Harriet Martin, reporting from Khartoum, said that the move could have a "huge impact on regional stability".
Both African countries agreed in March to resume crude exports from landlocked South Sudan through Sudanese oil facilities after resolving disputes over transit fees and other issues.
"If Sudan cuts the pipeline, it's going to be difficult to tell how the two sides will get back on track - not least because oil companies might just get fed up with pair of them," our correspondent said.
Bashir had warned at the end of May that he would block the oil if the South's government in Juba provides assistance to rebels fighting in South Kordofan and Blue Nile states, or in the Darfur region.
South Sudan's government denies supporting rebels in the north, and in turn has accused Khartoum of backing fighters on southern territory.