US President Donald Trump responded to North Korean leader Kim Jong-un's claim of having a nuclear button on his desk, saying that he too has one which is "much bigger" and "more powerful".

"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 and more powerful one than his, and my button works," Trump tweeted on Tuesday evening.

On Monday, Kim said in his annual New Year's Day address: "The entire mainland of the US is within the range of our nuclear weapons and the nuclear button is always on the desk of my office. They should accurately be aware that this is not a threat but a reality," CNN reported.

In the address, Kim also expressed a desire for a peaceful resolution with South Korea, a break from the aggressive language he used to threaten the US.

In a series of tweets on Tuesday morning, Trump said the potentially warm gesture to South Korea from Kim is "perhaps" good news, "perhaps not", and referred to "sanctions and 'other' pressures" on North Korea.

Later on Tuesday night, press secretary Sarah Sanders said at the White House news briefing that the US approach to North Korea had not changed, that the US continues to regard North Korea as a global threat and seeks an international solution while keeping "all of our options on the table".

Trump's statements came on the same day South Korea proposed to hold a high-level meeting with North Korea that could take place on January 9, reports Efe news.

This meeting, which Kim has not accepted yet, would be the first of its kind in more than two years between these two neighbouring countries.

Pyongyang's ongoing missile and nuclear tests in recent months have led the UN to adopt a series of economic sanctions against the regime.

However, Washington's response has been rather ambiguous.

While the State Department does not hesitate to find a diplomatic solutions for the conflict, Trump does not rule out a possible military action.