The South's 19th president - the son of refugees from North Korea - had during his campaign advocated for a combination of negotiations and economic cooperation alongside military and security measures in dealing with the North. Earlier in the day, Moon spoke with British Prime Minister Theresa May and German Chancellor Angela Merkel, the Blue House said.
On Friday, Russia's President Vladimir Putin told Moon in a phone call he was ready to play a "constructive role" in resolving North Korea's nuclear threat, South Korea's presidential office said.
Moon, who was elected this week, has taken a more conciliatory line than his conservative predecessors and has said he would be prepared to go to Pyongyang "if the conditions are right".
But until Thursday, South Korea had not been invited.
North Korea says it needs its weapons to defend itself against the United States which it says has pushed the region to the brink of nuclear war.
Taking the oath of office in Seoul on Wednesday, Moon said he will further bolster the alliance and "immediately fly to Washington if necessary" for the sake of peace on the Korean Peninsula.
"Threats from North Korea's nuclear and missile development have entered a new stage", Japan's Abe told Moon in their telephone call, according to Japanese Deputy Chief Cabinet Secretary Koichi Hagiuda.More news: Man United held by Southampton in 0-0 stalemate
China has protested against the deployment saying the system's powerful radar can probe deep into its territory, undermining is security, destabilizing the regional security balance and doing little to deter North Korea. Moon responded that he's aware of China's concerns and he hopes the two countries can have a greater understanding of each other's positions on the system, it said.
"I believe Russian Federation is our country's best partner in working to ensure the peace and prosperity of Eurasia", the new South Korean president told Putin, according to his presidential office Cheong Wa Dae.
As well as clouding efforts to rein in North Korea's nuclear ambitions, the THAAD deployment has also led to recriminations from Beijing against South Korean companies.
Moon will send a delegation headed by Park Byeong-seug, a veteran member of parliament and a senior official of Moon's liberal Democratic Party, Yoon said.
The last time that Han appeared to brief foreign diplomats in Pyongyang was last December, to present North Korea's response to the latest round of United Nations sanctions after the September 2016 nuclear test.
China has denied it is retaliating against South Korean businesses.