North Korea on Sunday launched the Hwasong-12, the projectile that has shown the best performance so far aside from its space rockets, in what the global community suspects to have been a covert missiles test. Upon receiving hand-written letter from Moon, Chinese President Xi Jinping said China was "willing to work with South Korea to preserve the hard-won results, properly handle disputes, and put bilateral ties back onto a normal track to benefits of both people on the basis of mutual understanding and respect".
North Korea launches four ballistic missiles by the Korean People's Army (KPA) during a military drill at an undisclosed location in North Korea.
The North has vowed to develop a missile mounted with a nuclear warhead that can strike the mainland United States, saying the program is necessary to counter US aggression.
"The reality is that there is a high possibility of a military conflict at the NLL (Northern Limit Line) and military demarcation line", Mr Moon was quoted as saying by the presidential Blue House.
While Trump might be tempted to consider some kind of surgical strike against North Korea's nuclear facilities, he could unleash a North Korean counter-attack against nearby Seoul with devastating destruction and countless deaths.
But he has said the North must change its attitude of insisting on pressing ahead with its arms development before dialogue is possible.
This is the first time Trump has met with a South Korean official since taking office.
However, experts say these pro-engagement policies are not likely to happen anytime soon, especially with the North's latest missile test on Sunday.
"We all have to send a sign to North Korea, and that is 'No more".
The North defends its weapons programmes as necessary to counter U.S. hostility and regularly threatens to destroy the United States.
Moon's envoy to the United States, South Korean media mogul Hong Seok-hyun, left for Washington on Wednesday.
Relations between Beijing and Seoul, strained by disagreement over South Korea's hosting of the U.S. Terminal High Altitude Area Defense ( THAAD ) system, have taken on a more conciliatory tone with the election earlier this month of President Moon Jae-in.
China has been infuriated by the United States deployment of an advanced Terminal High Altitude Area Defense (Thaad) anti-missile system in South Korea, saying it was a threat to its security and would do nothing to ease tensions with Pyongyang.
US Ambassador to the United Nations Nikki Haley has made it clear that Washington would talk to North Korea only after it halts its nuclear programme.
United States envoy to the UN, Nikki Haley, told reporters just prior to a UN Security Council meeting at which the UN is considering new sanctions against North Korea: "We are willing to talk but not until we see a total stop of the nuclear process and of any test there". Joseph Bermudez Jr. an analyst at the U.S. firm Allsource Analysis said to the Wall Street Journal: "What's more concerning is not an individual test or two individual tests, it's that they're approaching their missile development in a very pragmatic way".
Again, the olive branch Moon extended to President Kim came as a surprise, given North Korea's threats for decades to reduce Seoul, the capital of South Korea, to a "sea of fire".
South Korea and the U.S. say it is necessary to guard against North Korea's missile and nuclear weapons activities that threaten South Korea, Japan, U.S. territories and possibly even the continental United States.