In his meetings, Pompeo detected that the current leadership is prepared to make that "strategic shift".
Kim is the highest-ranking North Korean official to visit the country since 2000, when Jo Myong-rok, a vice marshal, visited Washington and met with U.S. President Bill Clinton to arrange a summit with leader Kim Jong-il.
The Journal quoted a foreign government source as saying the letter was "fairly basic".
Experts agree that a meeting between Kim and Trump would legitimize a pariah nation and grant North Korea a long-term wish. Immediately afterward, he tweeted that he had had substantive talks on the priorities for the potential summit. "A nuclear-capable North Korea with missiles that can reach the United States is unacceptable".
North Korea conducted a series of missile tests and nuclear detonations a year ago, which led to the imposition of additional United Nations Security Council resolutions meant to isolate the country economically.
Former four-star general and chairman of the Institute for the Study of War Jack Keane told Fox News that Trump would not engage in game-playing with his North Korean counterpart over their Singapore summit.
Kim Yong Chol speaking with Pompeo. There's a very good chance that it won't be done in one meeting or two meetings or three meetings.
Kim Yong Chol left his hotel in New York City early Friday for the trip to Washington in a convoy of SUVs.
'Come to Russia, we'll be very happy to see you, ' Lavrov told Kim in remarks released by the Russian foreign ministry. "It is very important to them".
On the possibility of a quick solution to the crisis, Trump said today as he was boarding Air Force One: "I'd like to see it done in one meeting".
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"As long as that is part of the discussion, we're going to continue to shoot for the June 12th and expect to do that".
Kim is carrying a letter for Trump from North Korean dictator Kim Jong Un.
"We made real progress in the last 72 hours towards setting conditions" needed to finalise plans for a Trump-Kim meeting, tentatively planned for June 12 in Singapore, Pompeo said in a briefing at the Lotte New York Palace hotel, part of the South Korean conglomerate Lotte Group.
Pompeo spoke after meeting with Kim Yong-chol for a little more than two hours at the residence of the deputy USA ambassador to the United Nations. "The people of #NorthKorea can have a brighter future and the world can be more peaceful".
He arrived in the US capital from NY, where he held talks with US Secretary of State Mike Pompeo on Thursday. In the meeting, Lavrov invited Kim Jong Un to visit Russia and meet with the Russian President Vladimir Putin.
The US side is pressing its demand for "complete, verifiable, irreversible denuclearization".
"I think we are looking for something historic", said the official, speaking anonymously to discuss the private meetings. Trump wants to use the meeting to pressure Pyongyang into giving up its nuclear weapons. North Korea's conciliatory response to that letter appears to have put the summit back on track.
North Korean vice chairman Kim Yong-chol arrived by auto at the formal diplomatic entrance to the White House on the South Lawn, a setting usually reserved for the arrival of important allies.
FILE - White House Press Secretary Sarah Sanders listens to a reporter's question during the daily briefing at the White House in Washington. However, these negotiations ultimately failed.