"I am concerned by reports that the clashes are continuing today with additional claims of new casualties," the UN official said.
"These potentially unlawful killings come just a month after at least 20,000 people were displaced by clashes in Las Anod, and could contribute to further displacement, compounding the already fragile humanitarian situation in the region," Turk said.
The UN rights chief called on authorities to conduct a "credible and impartial investigation" into the clashes.
Angry demonstrations broke out in the disputed border town last month, with opposition parties and rights groups accusing government forces of shooting dead several protesters.
The aim, he said, should be to "determine who is responsible and hold them to account in fair trials, including for reported damage to homes."