The top leader said he will "wait for a bold decision" from Trump "with patience till the end of this year".
"Our government will do what we can in order to maintain the current momentum for dialogue", Moon's office said in a statement.
North Korean leader Kim Jong Un said the breakdown in talks with the United States has raised the risks of reviving tensions, but he is only interested in meeting President Donald Trump again if the United States comes with the right attitude, state media KCNA said on Saturday.
At the end of the speech, Kim said he hoped that the United States would decide wisely and that the confrontation between the two countries will be brought to a complete stop.
Trump and Kim held their first landmark summit in Singapore last June, where the pair signed a vaguely-worded agreement on the "denuclearization of the Korean peninsula".
Kim said Friday that he's willing "try one more time" under the conditions that the USA "stop the current way of calculation" and approach talks with the "right attitude and (find) a methodology that can be shared with us".
Trump, however, said after the Hanoi summit that North Korea committed to "totally" dismantling its main Yongbyon nuclear complex, but the lifting of the sanctions would require Pyongyang to scrap other nuclear facilities and programs, including undeclared ones.
Negotiations for nuclear disarmament in Hanoi collapsed because the two leaders disagreed on sanctions and nuclear facilities, Mr. Trump told reporters.
But the talks broke down and the summit was unexpectedly cut short because North Korea wanted the United States to lift all sanctions, according to Trump.Читайте также: NBA Playoffs: Portland Trail Blazers vs. OKC Thunder preview
Kim complained that Washington is coercing South Korea into abiding by sanctions and not pushing forward with inter-Korea projects.
Speaking alongside South Korean President Moon Jae-in, Trump on Thursday expressed openness to "smaller deals" with the North. DPRK stands for the North's official name, Democratic People's Republic of Korea.
"The confrontation is likely to continue as there's a low chance that the USA will change its position", Mr Shin said.
The DPRK leader also accused Washington of "openly forcing the South Korean authorities to "control speed" and moving in every way to subordinate the implementation of the north-south agreement to its policy of sanctions and pressure on the DPRK".
On Friday, KCNA reported Kim was re-elected as chairman of the State Affairs Commission, the North's most important decision-making body, to cheers and loud applause from the delegates.
USA policy, he said, creates "deliberate hurdle in the improvement of ties while coming forward with unilateral gangster-like demands".
In his speech, Kim repeated earlier claims that North Korea's crippled economy would persevere through heavy global sanctions imposed over its nuclear weapons program and that he wouldn't "obsess over summitry with the United States out of thirst for sanctions relief".
"Choe Son Hui as first vice foreign minister puts her in position to be Biegun's counterpart if this goes back to the foreign ministry to negotiate", said Mr Victor Cha, a senior adviser at the Center for Strategic and International Studies in Washington.При любом использовании материалов сайта и дочерних проектов, гиперссылка на обязательна.
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