Not content to let North Korea's leader have the last word on nuclear capability, Donald Trump has fired off a bizarre tweet saying his button is "much bigger and more powerful".

During a televised New Year's Day speech, Kim Jong-un said the US would never be able to start a war with the rogue nation after it had developed the ability to hit all of the American mainland with its nuclear weapons.

"The entire United States is within range of our nuclear weapons, and a nuclear button is always on my desk. This is reality, not a threat," Kim said.

The US President hit back today, tweeting on Tuesday night: "North Korean Leader Kim Jong Un just stated that the "Nuclear Button is on his desk at all times." Will someone from his depleted and food starved regime please inform him that I too have a Nuclear Button, but it is a much bigger & more powerful one than his, and my Button works!"

North Korean Leader Kim Jong Un just stated that the “Nuclear Button is on his desk at all times.” Will someone from his depleted and food starved regime please inform him that I too have a Nuclear Button, but it is a much bigger & more powerful one than his, and my Button works!

— Donald J. Trump (@realDonaldTrump) January 3, 2018

The provocative tweet followed an earlier post during which Trump sounded open to the possibility of an inter-Korean dialogue.

During his New Year's address, Kim made a rare overture toward South Korea that he was willing to send a delegation to the Winter Olympics, which will be hosted by South Korea next month.

Trump alluded to his Kim's offer in a tweet this morning, which also referred to the dramatic escape of at least two North Korean soldiers across the heavily militarised border into South Korea.

"Soldiers are dangerously fleeing to South Korea. Rocket man now wants to talk to South Korea for first time. Perhaps that is good news, perhaps not — we will see!" Trump said, using his derisive moniker for the young North Korean leader.

Trump sent an earlier tweet saying the US-led campaign of sanctions and other pressure were beginning to have a "big impact" on North Korea.

In response to Kim's overture, South Korea on Tuesday offered high-level talks on January 9 at the shared border village of Panmunjom to discuss Olympic co-operation and how to improve overall ties.

But Trump's ambassador to the United Nations insisted talks would not be meaningful unless the North was getting rid of its nuclear weapons.

North Korea did not immediately react to the South's proposal. If there are talks, they would be the first formal dialogue between the Koreas since December 2015. Relations have plunged as the North has accelerated its nuclear and ballistic missile development that now poses a direct threat to America, South Korea's crucial ally.

The US administration, however, voiced suspicions that Kim was seeking to drive a wedge between Seoul and Washington. Pyongyang could view a closer relationship with Seoul has a way for reducing its growing international isolation and relief from sanctions that are starting to bite the North's meagre economy.

"We won't take any of the talks seriously if they don't do something to ban all nuclear weapons in North Korea," US Ambassador Nikki Haley told reporters at the United Nations. "We consider this to be a very reckless regime. We don't think we need a Band-Aid, and we don't think we need to smile and take a picture."

In his New Year's address, Kim repeated fiery nuclear threats against the United States. He said he has a "nuclear button" on his office desk and warned that "the whole territory of the U.S. is within the range of our nuclear strike."

North Korea has been punished with unprecedented sanctions at the UN over its weapons programs, and Haley warned Tuesday of more measures if the North conducts another missile test.

In Washington, State Department spokeswoman Heather Nauert did not express opposition Tuesday to South Korea holding talks with North Korea, but voiced deep scepticism about Kim's intentions, saying he may be "trying to drive a wedge of some sort" between the U.S. and its ally, which hosts 28,000 American forces.

South Korea's liberal President Moon Jae-in has supported Trump's pressure campaign against North Korea but he's less confrontational than the US president and favours dialogue to ease the North's nuclear threats. Moon has long said he sees the Pyeongchang Olympics as a chance to improve inter-Korean ties.

White House spokeswoman Sarah Huckabee Sanders said the US would continue to put "maximum pressure" on North Korea to give up its nukes. She added that South Korea shares that goal.