Trump said the relationship is not simply good, but great, and that their two presidential election victories should be celebrated together, according to the report.
Xi officially invited Moon to visit Beijing, Yoon added. The date was not set for a potential visit by Moon.
Conservative U.S. media voiced fears of potential friction between the two allies in dealing with a belligerent North Korea following nearly a decade of conservative rule in South Korea that led to a hard-line approach toward the North. China has vigorously opposed the deployment of a United States anti-missile system in South Korea, saying it threatens Chinese security and will do nothing to resolve tensions with North Korea. The usual circumstances of the election and immediate transition into office meant Moon inherited several officials from Park's government, and he has moved quickly to replace them.
"Putin and Moon agreed on further contacts", the Kremlin said. "The president said he will send a special envoy in the phone conversation, so we will be ready to carry out preparations as soon as he calls for it".
Moon assumed presidential duties early in the morning after the National Election Commission finished counting Tuesday's votes and declared him victor of the special election necessitated by the ousting of conservative Park Geun-hye, whose downfall and jailing on corruption charges is one of the most turbulent stretches in the nation's recent political history. He is reportedly against the aggressive military measures Trump has threatened to deploy if North Korea does not back down from vigorously expanding its nuclear regime.
He also said he was prepared to go to Pyongyang "if the conditions are right".
The two leaders may not see eye-to-eye on what could comprise the right circumstances.
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Moon and Trump may also disagree on the US missile defense system THAAD. The leaders agreed to strengthen the alliance.
Tension has been high for months on the Korean peninsula over North Korea's nuclear and missile development and due to fears it will conduct a sixth nuclear test and more ballistic missile launches in defiance of UN Security Council resolutions.
In his inaugural address, Moon said his government "will negotiate with China to resolve the issue over THAAD". What decision Moon makes on THAAD's deployment will be keenly watched in the region and beyond.
Moon also had a telephone call with Japanese Prime Minister Shinzo Abe, Japanese news agency Jiji reported.
Moon did not express a negative view of the 2015 "comfort women" agreement, the official told reporters.
Suga only said Abe told Moon that the agreement should be "appropriately managed", but did not elaborate.
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