The Rodong Sinmun said a "fundamental shift" in policy was needed to reassure North Korea of Moon's intentions.
If the government meeting goes ahead, it will mark the first official inter-Korea talks since December 2015.
The Japanese press report also stated President Barack Obama relayed the administration's view on North Korea regime collapse to then president-elect Donald Trump, during their first meeting in November, following Trump's election victory.
Much ink has been spilled as to what this test means for the US, how policymakers should address North Korea nuclearization efforts, and the likelihood of war between the two nations.
South Korea also proposed separate talks by the rival states' Red Cross organisations to resume a humanitarian project to reunite families separated during the 1950-53 Korean War.
In April, China positioned 150,000 troops along their much larger border with North Korea last week, as well as sending a fleet of fully-loaded cargo ships destined for North Korea back home.
Those talks would be held on August 1, also in Panmunjom - that is, if Pyongyang accepts.
The bill would ensure companies working with North Korea are locked out of the American financial system by making the White House "block all transactions that are property of the North Korean government, affiliates, or those that do significant business with North Korea", "block any entity or financial institution implicated in any significant trade in goods or services with North Korea from the US financial system" and "prohibit any goods made with North Korean labor from entering the United States and to impose sanctions of all entities implicated in North Korean labor trafficking".
Monday's European Union talks were held amid mounting tensions over North Korea's nuclear program, following a string of missile tests in recent months that showed new capabilities.
The proposals come after Moon said at the G20 summit in Hamburg this month he was in favour of dialogue despite the "nuclear provocation" of North Korea's latest missile test.
In the proposal for talks, South Korea did not elaborate on the meaning of hostile military activities, which varies between the two Koreas. "We absolutely disagree on that", he said.
What is more urgently concerning is the fact that North Korea already has all of the technology it needs to start a damaging war with the USA via its short- and medium-range missiles.
When Moon visited Washington after being elected president, he and Trump said they were open to renewed dialogue with North Korea but only under circumstances that would lead to Pyongyang giving up its weapons programmes. -South Korea military drills.
Pyongyang has said it will not engage in talks with Seoul unless it turns over 12 waitresses who defected to South Korea past year after leaving a North Korean restaurant in China.
The two Koreas restarted Cold War-era psychological warfare after the North's fourth nuclear test in January 2016.
Cho stressed that Seoul "would not seek collapse of the North or unification through absorbing the North".