But others have been more cautious, arguing that the Moon administration should first develop a foreign policy framework that covers THAAD, defense cost-sharing, the free trade agreement (KORUS FTA) and North Korea and its nuclear program to prepare to negotiate with Trump, given his reputation for making a hard sell.
The two leaders said they looked forward to meeting at the Group of 20 summit meeting in Germany in July.
Park, who also served as co-chairman of Moon's presidential campaign team, is expected to meet Chinese officials on the sidelines of the forum to discuss the ways to fix relations, the official told Yonhap.
China's foreign ministry made no direct mention of the anti-missile system in its statement about the discussions.
Moon said he planned to send a special envoy to Russian Federation in the near future.
A high-ranking Japanese official, speaking on condition of anonymity, said Moon's willingness to meet Abe marks "a clear difference" from the attitude of former South Korean President Park Geun-hye. USA officials have said they see no value in resuming global talks with North Korea until Pyongyang has made clear it is committed to denuclearization.
The two leaders may not see eye-to-eye on what could comprise the right circumstances.
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.
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Mr Moon held the 40-minute phone call with his Chinese counterpart at the Blue House, the presidential residence.
The two sides agreed on the importance of denuclearizing the peninsula.
While Moon may try to soothe Chinese anger over the missile-defense system, many observers say it appears too late to call for its withdrawal, which would greatly undermine South Korea's relationship with Washington.
At home, Mr Moon will have to deal with slowing growth, soaring unemployment and public frustration over widening gaps in wealth and opportunities.
Moon, who took office on Wednesday, favors engagement with the North - whose key diplomatic backer is China - to bring it to the negotiating table over its nuclear and missile ambitions.
Moon acknowledged China's concerns over the deployment of the Thaad system, but went on to request Xi to take a special interest in the concerns of Korean citizens and businesses in China that are suffering from a backlash over the Thaad deployment, asking for the restrictions and sanctions to be "resolved amicably".
During his call with Japanese Prime Minster Shinzo Abe, the two leaders agreed that their countries must not let their hard history hamper co-operation in dealing with North Korea's nuclear program, Moon's office said.
Relations between Beijing and Seoul have deteriorated over the issue.
He touched upon the 2015 agreement reached by Japan and South Korea to resolve the issue of "comfort women" who were forced to provide sex to Japanese soldiers before and during World War II.