In his interview with Reuters, the USA president said a nuclear deal with North Korea would have to cover its missile program.
US -ally South Korea and North Korea have technically been at war for decades, even though the Korean War's military combat ended in 1953, because a peace agreement was never signed.
Kim's visit represented an extraordinary turn of events.
Meanwhile, President Trump will also meet Japan's Prime Minister Shinzo Abe in Washington on June 7.
Trump's announcement came at the end of a two-hour meeting in the Oval Office of the White House with North Korean envoy Kim Yong Chol, who had travelled from NY after two days of talks with US Secretary of State Mike Pompeo. "Our secretary of state is having very good meetings".
"I look forward to the day I can take the sanctions off North Korea", Trump said.
Kim Yong-chul arrived at the White House Friday afternoon after two days of talks with Secretary of State Mike Pompeo, bearing a letter for Mr. Trump from Mr. Kim.
In the latest sign of hostility cooling down but hopes kept in check, Trump said he's unilaterally put a hold on new sanctions against the North "until the talks break down".
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McDowell County Emergency Management deputy director Adrienne Jones says the dam hasn't failed, but inspectors are watching it. Forecasters warn storms could bring more flooding Wednesday, but say the heavy rains will be isolated and not widespread.
Summit hangs on Kim letter after nuclear talks make progress
He is scheduled to meet with Secretary of State Mike Pompeo to discuss a prospective Trump-Kim meeting, according to Sanders. Yonhap said Kim, who had arrived in Beijing on Tuesday, was travelling with five other North Korean officials.
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He snubbed France, Germany and Britain by pulling out of a nuclear agreement with Iran and upset the Europeans, as well as neighbors Canada and Mexico, with protectionist trade policies meant to safeguard U.S.jobs.
U.S. Secretary of State Mike Pompeo met Kim's right-hand man, General Kim Yong Chol, in NY late on May 30 to discuss the matter.
In Seoul, US negotiators on Friday expressed optimism after meeting their North Korean counterparts for preparatory talks at Panmunjom, on the fortified border between the two Koreas. But in March, the bellicose rhetoric gave way to a proposal for a historic summit.
Pompeo touts "real progress" toward Korea summit State Dept releases global cyber recommendations for Trump MORE in NY this week.
"I'm still convinced that if he's in this process, even if the summit is ongoing and we're moving out, if this thing isn't heading towards the obvious objective he'll pull out".
Mr Trump has refused to publicly acknowledge whether he's spoken directly with Kim Jong Un ahead of the talks.
The Trump administration has pressed for the North to rapidly takes steps to turn over its nuclear program, while Pyongyang has said it expects a slower, step-by-step process in which the North receives reciprocal benefits, including economic incentives. He has been granted special permission for official travel to the United States.
Lavrov's May 31 visit - his first to North Korea since 2009 - was seen as an attempt by Moscow to ensure its voice is heard in Pyongyang's diplomatic overtures with the United States and South Korea.
He is also linked to the cyber-attack on Sony Pictures in 2014 and led the country's delegation to the 2018 Winter Olympics closing ceremony in South Korea's Pyeongchang earlier this year.
The White House said talks of the "total denuclearization of the peninsula" do not extend to USA weapons systems - a defense umbrella covering South Korea that includes nuclear-armed submarines and strategic bombers capable of carrying nuclear bombs not based on the peninsula. USA teams also met with North Korean delegations in Singapore and the Korean demilitarized zone.