Speaking outside the White House after briefing Trump, South Korean National Security Adviser Chung Eui-yong said he had passed on a message that Chairman Kim was "committed to denuclearisation" and had "pledged that North Korea will refrain from any further nuclear or missile tests".
The White House said Trump had agreed to the proposed face-to-face meeting. "He will accept the invitation to meet with Kim Jong-Un at a place and time to be determined," said White House spokeswoman Sarah Sanders.
If the meeting takes place, it would be the first ever between leaders of the two countries.
Trump himself confirmed the meeting in a tweet, adding that US sanctions would remain in place until a denuclearisation deal was achieved.
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"Kim Jong-Un talked about denuclearisation with the South Korean Representatives, not just a freeze. Also, no missile testing by North Korea during this period of time. Great progress being made but sanctions will remain until an agreement is reached. Meeting being planned!," Trump tweeted.
North Korea last year increased the pace of its missile programme. On November 29 last year, the North Korean leader said that his country had achieved full nuclear statehood after what he said was the successful test of a new missile capable of striking anywhere in the United States.
The two nations had recently threatened to wipe each other out.
Trump's decision to meet Kim, after a year in which the two have repeatedly traded insults, is being seen as a remarkable breakthrough.
South Korean National Security Adviser Chung Eui-Yong, who led his country's delegation to Pyongyang this week for talks with the North Korean leaders, briefed Trump and his national security team.
Reading from a prepared statement, Chung said the North Korean leader has expressed his "eagerness to meet with President Trump as soon as possible."
In his meeting with the North Korean leader, Chung said Trump is committed to denuclearisation of the Korean Peninsula.
"Kim pledged that North Korea will refrain from any further nuclear or missile tests. He understands that the routine joint military exercises between the Republic of Korea and the United States must continue. He expressed his eagerness to meet President Trump as soon as possible," he said.
"The Republic of Korea, the United States, and our partners stand together in insisting that we not repeat the mistakes of the past, and that the pressure will continue until North Korea matches its words with concrete actions," the South Korean national security adviser said.
Congressman Ed Royce, Chairman of the House Foreign Affairs Committee, said Kim's desire to talk shows sanctions the administration has implemented are starting to work.
"We can pursue more diplomacy, as we keep applying pressure ounce by ounce. Remember, North Korean regimes have repeatedly used talks and empty promises to extract concessions and buy time," he said.
Trump's decision is a remarkable breakthrough. It brings the North Korean regime close to its long-desired aim of recognition on the international stage, and offers Trump the tantalising prospect of a historic diplomatic victory. But the consequences of such a high-stakes gamble remain hard to predict, CNN commented.