The incident occurred on Tuesday at 01:44 p. m. local time (04:44 GMT).
"Now it is being determined whether the pilot has ejected," the military said.
The South Korean Yonhap news agency reported that the fighter jet crashed into a mountain in Hwaseong, Gyeonggi province. The cause of the accident is yet to be determined.
The Northrop F-5 is a family of supersonic light fighter aircraft designed in the late 1950s by the US aerospace and defence Northrop Grumman Corporation.