Pompeo is meeting with the Japanese and South Korean foreign ministers in Seoul.
"I would like to re-extend, and ask you to convey to the leader of North Korea Kim Jong Un, our invitation to visit Russia", Putin told Kim Yong-nam, chairman of the presidium of the Supreme People's Assembly of North Korea, according to a transcript by the Kremlin.
The Russian leader praised the historic meeting between the United States and North Korea, adding it was "without doubt just the first step towards a full-blown settlement".
He said the meeting helped "push back that unfavourable scenario and raised prospects for solving all the problems with peaceful and diplomatic means".
The comment came as Secretary of State Mike Pompeo conveyed the U.S. resolve for "complete, verifiable and irreversible denuclearisation" of the Korean Peninsula with Chinese President Xi Jinping, the State Department said, as well as concerns about Chinese activities in the South China Sea. "We are going to get complete denuclearisation; only then will there be relief from the sanctions". The summit in Singapore marked a reduction in tensions-a sea change from last fall, when North Korea was conducting nuclear and missile tests, and Trump and Kim were trading threats and insults that stoked fears of war.
Trump has also been widely criticized for his friendliness toward North Korean leader Kim Jong-un during and after the summit.
The resolutions "have mechanisms for relief contained in them and we agreed that at the appropriate time that those would be considered", Pompeo said.
The president's suspension of the exercises came as U.S. Forces-Korea, led by Army Gen. Vincent Brooks, is preparing for the annual Ulchi-Freedom Guardian exercises later this summer with the South Koreans.
The United States did, however, commit to suspending joint military exercises with South Korea, Trump told reporters before departing the summit site.
U.S. Secretary of State Mike Pompeo says President Donald Trump's comment about North Korea no longer posing a nuclear threat was made "with eyes wide open".
Seeking to quell concerns that North Korea and the USA interpreted the talks differently, Pompeo said, "I am confident that [North Koreans] understand what we're prepared to do".
"You could argue semantics, but let me assure you that it's in the document", Pompeo said.
Even as experts weighed the implications of Trump's Singapore summit with Kim, the USA president struck a typically bullish note in a series of announcements. North Korea reaffirmed previous promises to work towards the "complete denuclearisation of the Korean peninsula". Trump reportedly had been briefed on saluting protocol, which is not to salute military officers from other countries, but rather to shake hands after their salute.
The KCNA report - which was also carried by the Rodong Sinmun, official newspaper of the ruling Workers' Party - also claimed that the president pledged to suspend military drills with South Korea, and lift sanctions on the North.
Trump also said he hopes the thaw in relations with North Korea will eventually allow him to withdraw US troops from the peninsula. But Lu Chao, an expert on North Korea at the Liaoning Academy of Social Sciences in northeast China's Shenyang city, says the guarantees are probably "a repeat of previous US statements that it does not seek to overthrow the current regime in North Korea".