Chinese and North Korean state media reported earlier Thursday that Kim told Xi that he's committed to setting up a second summit with Trump to "achieve results" on the nuclear standoff on the Korean Peninsula.
The leaders were seen to have discussed North Korea's plans for a second summit with U.S. President Trump.
At their first meeting in Singapore in June, Kim and Trump signed a vaguely worded document with Kim pledging to work towards the "denuclearization of the Korean peninsula".
Moon added that matters related to establishing peace and declaring the end of the Korean War can be considered as measures corresponding to Pyongyang's denuclearization steps, and that he hopes the differences in Pyongyang and Washington's views on the issues will be resolved through the second Trump-Kim meeting.
Kim, for his part, welcomed China's readiness to "jointly study and steer" the situation to help ease tensions, as well as the positive role Beijing continues to play in the negotiations process, KCNA said.
Moon's overture, however, has invited criticism from conservatives in South Korea and the United States that he's making too many concessions and helping the North weaken US -led sanctions. "If he wants to shift his focus to the development of the economy, he needs China's cooperation", said Lu Chao, a North Korea expert at China's Liaoning Academy of Social Sciences.
By inviting Kim to meet him in China before a summit with Trump, Xi may be sending a message to Washington that "Beijing still has leverage over Pyongyang", according to Lim.
North Korea wants relief from the multiple sets of sanctions imposed on it over its nuclear weapons and ballistic missile programmes, while the USA wants the measures to remain in place until it gives up its arms - something Pyongyang has made no public promise to do.
Confusion as Malaysia king abdicates throne after marrying Russian
He has also expressed his highest appreciation to the prime minister and the government for their cooperation, the statement said. The report also said that Sultan Muhammad was anxious to start a family after he did not have children in his first marriage.
South Korea's Yonhap News Agency said Mr. Kim visited a technology development zone and spent around 20-30 minutes touring a factory run by famed traditional Chinese medicine maker Tong Ren Tang.
China, which wouldn't want to be left out in the cold in any settlement, has worked to improve relations with its neighbour, which deteriorated after Beijing backed United Nations sanctions against North Korea.
"I expect we'll hear about high-level talks between North Korea and the United States before long for their second summit", he added.
Each of the previous Kim-Xi meetings have come shortly before or after the North Korean's summits with either Trump or South Korean President Moon Jae-in.
North Korean leader Kim Jong-un was given a lavish welcome in the Chinese capital on Tuesday, which was his birthday.
Some analysts say China could use its cooperation on the North Korean issue as a bargaining chip in the USA trade talks. Kim was accompanied by his wife, Ri Sol Ju, the news agency said.
Moon, however, made it clear that North Korea taking denuclearization steps is the key to resolving the matter.
"China has been an important force in promoting the political settlement of the Korean Peninsula nuclear issue and peace and stability on the peninsula", said Lu.
However, a meeting between Mr Pompeo and senior North Korean official Kim Yong Chol has yet to be rescheduled after an abrupt cancellation in November.