The Demilitarised Zone is a strip around 2.5 miles wide that runs across the entire width of the Korean Peninsula, separating the North and South. The military threat is the latest in a string of increasingly hostile statements made by North Korea in recent weeks.
Tensions have reached a new high as the North continues to crack down on the South.
The two countries previously reached a tentative peace deal during talks in 2018.
However, the North is now thought to be applying pressure in the hope of securing relief from crippling international economic sanctions.
Pyongyang has also accused the South of failing to prevent defectors from sending anti-regime material back over the border.
The Demilitarised Zone has been in place since 1953 and marked the end of hostilities during the Korean War.
The zone continues to host diplomatic meetings and military negotiations between the two countries.
The General Staff of the Korean People’s Army, the armed forces of North Korea, laid out the latest threats in a statement carried by the state-run Korean Central News Agency.
“Our army is keeping a close watch on the current situation in which the north-south relations are turning worse and worse,” it read.
JUST IN: North Korea urged to rein in rhetoric after military threats
On Monday, South Korean president Moon Jae-in has called on the North to cease the escalation of tensions.
“We must not push back the pledges of peace that Chairman Kim Jong Un and I made,” he said in comments released by his office.
Meanwhile, North Korean leader Kim Jong-Un has been urged stop stoking tensions and return to dialogue with South Korea as the spectre of military conflict returned to the region.
Pyongyang has threatened to cut all ties with Seoul and also warned it is prepared take military action against its neighbour as relations nosedive.
South Korean President Moon Jae-in called on Kim to refrain from adding to the renewed tensions and return to dialogue.
President Moon said: “The promises of peace on the Korean peninsula that Chairman Kim Jong-un made before our 80 million people cannot be turned back.
“North Korea should not cut communications, raise tension and try to go back to the past era of confrontation.
“I hope it will resolve uncomfortable, difficult problems through communications and cooperation.”
The president’s remarks came as South Korea marked the 20th anniversary of the first summit between the two nations which pledged to step up dialogue and cooperation.
At their own first summit in 2018, Mr Moon and Kim signed a declaration to work for a “complete denuclearisation of the Korean peninsula” and cease “all hostile acts”.
But the North cut hotlines with its neighbour last week and vowed to suspend all contact if Seoul did not take action to stop the defectors’s groups from sending sending leaflets and other material.
In a separate anniversary message, Mr Moon said: “I also regret that North Korea-US and inter-Korean relations have not made progress as expected.
“But what’s most important is trust, which the South and North should build through constant dialogue.”