President Omar al Bashir issued a resolution declaring the emergency in the border states of South Kordofan, White Nile and Sennar, according to the official SUNA news agency.
The measure suspends the constitution and imposes a trade embargo against the South, which became independent last year. It also gives the authorities wide powers of arrest.
Nationalist feeling has intensified in Sudan after South Sudan occupied its main Heglig oil field for 10 days in a move that coincided with Sudanese air strikes against the South.
The Sudanese army has claimed that four people arrested in Heglig, including a Briton, Norwegian, a South African and a South Sudanese, have military backgrounds.
But Ashley Williams, CEO of Mechem which employs the South African and South Sudanese people now in custody, insisted they were not involved in the "aggression" in the region.
"It's humanitarian work so the story of them being military advisers and this type of thing is completely and utterly nonsense and not true," Mr Williams said.
"The abduction took place well within South Sudan territory," as they were travelling south towards their base in a mine-protected vehicle, he said.
"Then they grabbed them and drove back to Heglig with them where they then said they've arrested them in this disputed area while they weren't there at all."
Al Obeid Meruh from Sudan's foreign ministry said the diplomats' request for access has been transferred "to the authorised agency".
The foreigners crossed illegally into Sudan and "we don't care if they are from the UN or not," he said, adding: "Until now those people are under investigation."
A British embassy spokeswoman said officials are "urgently investigating the arrest of a British national in Sudan" and had requested consular access.
South Sudan broke away from Sudan in July last year after a peace deal ended one of Africa's longest civil wars, which killed about two million people between 1983 and 2005.
Tensions have risen over a series of unresolved issues including the border, the future of disputed territories and oil.
South Sudan's embassy estimates 350,000 ethnic Southerners remain in the north after an April 8 deadline for them to either formalise their status or leave Sudan.
Twelve thousand are camped south of Khartoum at the way-station of Kosti, awaiting a trip home by river barge.
But the governor of White Nile state, which includes Kosti, has branded them a security threat and given them a May 5 deadline to get out, official media reported on Sunday.