Peskov refused to say what was in the message during a conference call with reporters on Thursday, and declined to comment on the prospects for a possible meeting between Putin and Kim.
But the presidential office remains cautious, saying that holding a three-way summit and declaring an end to the Korean War all depends on the outcome of the North Korea-US summit.
But Pompeo suggested things are moving in the right direction.
But the White House says talks of the "total denuclearization of the peninsula" do not extend to US weapons systems - a defense umbrella covering South Korea which includes nuclear-armed submarines and strategic bombers capable of carrying nuclear bombs not based on the peninsula. "So far, the readout from these meetings has been positive, and we'll continue to move forward in them".
"President Trump and I believe Chairman Kim is the kind of leader who can make those kind of decisions, and in the coming weeks and months, we will have the opportunity to test whether or not this is the case".
A pair of US delegations have held talks this week with their North Korean counterparts at the Korean demilitarized zone and in Singapore. There's a very good chance that it won't be done in one meeting or two meetings or three meetings.
"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.
North Korea conducted a series of missile tests and nuclear detonations previous year, which led to the imposition of additional United Nations Security Council resolutions meant to isolate the country economically. Still, Pompeo said he doesn't yet know whether the meeting will proceed on the announced date of June 12 and that he believes North Korea's leaders are contemplating a different path forward that would allow their nation to more fully integrate into the global community.
The usually secretive Kim Jong-un has given a glimpse behind North Korea's political veil by allowing news cameras inside the nation's parliamentary building during his historic meeting with Russian Foreign Minister Sergei Lavrov.
LeBron James left in disbelief after stellar Game 1 performance
The top scorers for Golden State were Steph Curry with 26 points, Kevin Durant with 26 and Klay Thompson with 24. Coach Steve Kerr admitted they "got lucky" on Smith's goof-up and "got a break" from the refs.
Washington wants North Korea to quickly give up all its nuclear weapons in a verifiable way in return for sanctions and economic relief.
Privately, there is increasing acknowledgment in the administration that any disarmament will not be immediate but could take years, and will require U.S. concessions along the way - a step-by-step process that some senior White House aides have previously rejected, citing North Korea's history of reneging on its promises.
Since time is of the essence, American officials have held preparatory meetings with North Korean delegations in both Singapore and the demilitarized zone, the border area separating North and South Korea, in the past few days. But his comments Thursday come at a sensitive moment, when a senior North Korean official was in NY for talks with US Secretary of State Mike Pompeo on the planned summit.
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. The White House said the were scheduled to have a full day of talks on Thursday.
"Very good meetings with North Korea", Trump tweeted of the talks.
An aide to Kim Jong Un has arrived at the White House, becoming the highest-ranking North Korean official to visit in 18 years.
In 2016, he took charge of the United Front Department, the civilian intelligence agency which supposedly operates pro-North Korean groups in South Korea and handles inter-Korean affairs.
Russian Federation is the latest major nation to reach out to North Korea since Trump accepted Kim's proposal for a summit.