The statement comes amid a deadlock in nuclear negotiations between the United States and North Korea.
North Korea says it will never unilaterally give up its nuclear weapons unless the United States removes its nuclear threat first.
The statement carried by the official Korean Central News Agency on Thursday comes amid a deadlock in nuclear negotiations between the United States and North Korea.
It raises further doubts on whether North Korean leader Kim Jong-un will ever relinquish an arsenal he may see as his greatest guarantee of survival.
Kim and President Donald Trump met on June 12 in Singapore where they issued a vague goal for the "complete denuclearisation" of the Korean Peninsula without describing when and how it would occur.
But North Korea for decades has been pushing a concept of denuclearization that bears no resemblance to the American definition, vowing to pursue nuclear development until the United States removes its troops and the nuclear umbrella defending South Korea and Japan.
'Accurate meaning of denuclearisation'
In Thursday's statement, the North reiterated its traditional stance on denuclearisation and accused Washington of misleading what had been agreed on in Singapore.
"The United States must now recognize the accurate meaning of the denuclearisation of the Korean Peninsula, and specially, must study geography the right way," the statement said.
"When we talk about the Korean Peninsula, it includes the territory of our republic and also the entire region of (South Korea) where the United States has placed its invasive force, including nuclear weapons. When we talk about the complete denuclearization of the Korean Peninsula, it means the removal of all sources of nuclear threat, not only from the South and North but also from areas neighboring the Korean Peninsula," the statement said.
The US removed its tactical nuclear weapons from South Korea in the 1990s.
The nuclear negotiations between Washington and Pyongyang stalled since the Trump-Kim meeting.
The United States wants North Korea to provide a detailed account of nuclear and missile facilities that would be inspected and dismantled under a potential deal, while the North is insisting that sanctions be lifted first.
The North Korean statement came a day after Stephen Biegun, the Trump's administration's special envoy on North Korea, told reporters in South Korea that Washington was reviewing easing travel restrictions on North Korea to facilitate humanitarian shipments to help resolve the impasse in nuclear negotiations.