Relevant authorities were still verifying the facts, the statement added.
The JCS said troops were dispatched but they were unable to find the person, who crossed the Military Demarcation Line into North Korea at around 10:40 p.m.
A military official said security cameras had earlier captured the person climbing a barbed wire fence at around 6:40 p.m. Saturday, but the guard on CCTV duty had missed it.
The official said they were currently working to identify the person and that it is unknown whether the person was still alive.
“We sent a notice to North Korea to protect our people yesterday and there has been no response from North Korea,” South Korea’s Defense Ministry spokesman Boo Seung-chan said Monday.
There were “no unusual movements of the North Korean military” in relation to Sunday’s incident, he added.
North Korean state media did not report on any crossings over the DMZ overnight.