Moon's push to improve frayed ties hasn't reported much progress, with the North test-launching a series of newly built missiles.
The tested engine can be potentially used on intercontinental ballistic missiles, Fox News reported citing USA officials.
US President Trump said earlier this year that he would be open to talking with North Korean leader Kim Jong-un one-on-one under the right circumstances, but White House spokesman Sean Spicer poured cold water on that prospect on Tuesday in light of Mr Warmbier's death.
"The army of the DPRK is whetting the sword of retaliation shaper [sic] than before with a firm hold on the nuclear sword of justice to cope with the US imperialists' escalating aggression moves", North's Korean state-run Korean Central News Agency said on its website.
South Korea's new President Moon Jae-in said Seoul has no plans to scale back joint military exercises with Washington, according to an interview with United States broadcaster CBS.
Meantime, rhetoric coming from North Korean state media Thursday appeared to indicate Pyongyang is pressing forward with its nuclear weapons and missile development.
Test likely meant to underscore South Korea's capability to respond to a North Korean missile attackOn June 23, President Moon Jae-in visited the test site of the Agency for Defense Development, located in Taean, South Chungcheong Province, to observe the test launch of the Hyunmoo 2 ballistic missile. The North's sole International Olympic Committee member, Chang Ung, is travelling with the taekwondo team and will likely meet South Korean officials over cooperation on next year's Winter Olympics in Pyeongchang, South Korea.More news: Tottenham Flop Paulinho Confirms Offer From Barcelona
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President Trump has repeatedly said China is the key to reining in Pyongyang, but on Tuesday tweeted that "it has not worked out".
Just two days earlier, Trump had condemned the "brutality of the North Korean regime" after the death of Otto Warmbier, an American student who was detained in North Korea for almost a year and a half.
Mark Fitzpatrick, International Strategic Studies says, "I Am pretty sure that North Korea would be ready to test a nuclear weapon nearly whenever they make a political decision to do so".
In such a case, there will be a breakthrough in the impasse over North Korea's nuclear program, he added.
In a statement carried by South Korea's Yonhap News Agency, the JCS said that the soldier had crossed at the "central forward area" of the MDL at 21:30 KST on Friday.