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There are reports that Trump has already agreed to meet Kim by May.
The possible get-together is apparently being brokered by South Korea.
After a year in which North Korea fired intercontinental ballistic missiles capable of reaching all of the United States and tested what is widely thought to have been a hydrogen bomb, any break in the tensions between the two countries would likely be welcomed around the world.
There has never been a face-to-face meeting, or even a phone call, between the sitting leaders of the two countries.
The leader of the South Korean delegation, the country's national security adviser Chung Eui-yong, said at the White House that Kim was committed to demilitarisation, Trump called Kim "little rocket man" after North Korea conducted a series of ballistic missile tests.
A South Korean delegation reportedly delivered the letter offering a meeting to Trump at the White House.
Trump has previously indicated he was open to dialogue with North Korea, but the US administration has said North Korea must first take tangible steps towards denuclearisation.
But Trump has also said the US would rain "fire and fury" on North Korea.
US Secretary of State Rex Tillerson said yesterday in Ethiopia that the US has seen "potentially positive signals'' from North Korea, but the adversaries are still a long way from holding negotiations.