The South Korean President said he was briefed on Trump's recent comments, where he said that he had received another "beautiful" and "warm" letter from Kim.
The letter was apparently sent on the occasion of the first anniversary of the first summit between the two leaders in Singapore.
"There's a very interesting part that President Trump did not announce," Moon said during a joint press conference with Norwegian Prime Minister Erna Solberg.
Moon reiterated that he was ready to meet Kim at any time. "Accordingly, the timing is up to Chairman Kim," he said.
Both Moon and Kim had met thrice last year, with the first meeting being held after a gap of 11 years.
Talking about inter-Korean relations, Moon underlined that "substantive progress" in denuclearisation is the key for achieving peace and stability in the Korean Peninsula despite the easing of military tensions between the two sides.
"The Demilitarized Zone dividing the Korean Peninsula is being transformed into a peace zone. At the same time, humanitarian exchanges and support will continue," he said.
Backing the lifting of international sanctions on North Korea, Moon said that inter-Korean ties could be strengthened further with the initiation of various economic cooperation projects, including the reopening of the Kaesong Industrial Complex.
Meanwhile, Solberg said that North Korea's nuclear weapons programmes are a potential "destabilizing factor" in regional and global security.
"That's why security situations on the Korean Peninsula are a focus of global attention," she said.
The Norwegian Prime Minister also said that she and Moon noted that denuclearisation is an "extremely important" goal not only in the region but also globally.
North Korea has repeatedly insisted that the removal of sanctions will help spur economic growth, whereas the US has reaffirmed that sanctions will not be removed until the communist country completely stopped its nuclear weapons programme.