Witnesses to the conflict told the independent Sudan Tribune that fighting among three rival tribes left more than 12 people dead.
The report adds that warring factions have been at odds over rights to land use in the South Kordofan since last year.
All parties to the conflict were called on by a state security committee to "to demonstrate the highest degree of caution, patience and wisdom to overcome the events."
Last week, South Sudanese President Salva Kiir met with Sudanese President Omar al-Bashir in Ethiopia to discuss regional security issues. A 2005 peace deal that paved the way to South Sudan's independence in 2011 is threatened by disputes over oil, border clashes and ethnic conflict.
African Union negotiator Thabo Mbeki said last week that both sides agreed to set up a demilitarized zone along their shared borders, the BBC reports.
A peace document reviewed by the BBC said African negotiators were working on the timetable for implementing recent agreements.