South Kordofan Gov. Ahmed Haroun was recorded on film telling Sudanese forces that bringing rebels back alive would be "an administrative burden."
"Hand over the place clean, swept, rubbed and crushed," he was overhead as saying.
U.N. High Commissioner for Human Rights Navi Pillay warned that Haroun's statements could be considered a crime.
"We urge all parties to the conflict to refrain from attacks against civilians and to refrain from making inflammatory statements that could amount to incitement to commit gross violations of human rights and humanitarian law," she said in a statement.
Haroun is wanted by the International Criminal Court for his role in the genocide in Darfur, where he once served as governor.
South Sudan split off from Sudan in July as part of a comprehensive peace agreement reached in 2005. Border conflicts, ethnic tensions and disputes over oil threaten to unravel the truce, however.
Both sides blame the other for the violence. The governments had agreed to the terms of a non-aggression pact this week, though South Sudan walked away after rejecting claims it was backing rebel groups fighting Sudanese forces in the border states of South Kordofan and Blue Nile.
Sudanese officials defended Haroun's statements by saying he was trying to boost troop morale.