Trump has said the US and South Korea should stop their joint military exercises as long as both sides are negotiating in good faith, which the president says is happening.
"It will lead other dictators to think that however small a country they might run, they only need to focus on quickly building a nuclear weapons arsenal so that they can be an equal to the US", he said.
Tuesday saw the South Korean Presidential office say Seoul "needs to figure out the precise meaning and intention of Trump's comments".
On the other hand, some said that while they admit the efforts made by President Trump have contributed to the realization of the extraordinary summit, they fear that Kim and Trump are figures who are "hard to predict".
The Kim-Trump summit meeting centred on nuclear disarmament and reducing tensions concluded with a one-page agreement.
From 1950 to 1953 the Korean War waged on. "I do not know with whom the North Korean leader is negotiating".
Foreign Minister Julie Bishop said the focus was now on North Korea's response.
The VFW sent a June 6 letter to President Trump to urge him to make the issue part of the negotiation.
The presidential office said the measures were particularly needed at a time "when a variety of serious dialogues have been held between the North and the USA for the denuclearization of the Korean peninsula and the establishment of relations".More news: Kate Middleton takes George and Charlotte to watch William in polo match
Trump also said he wants "at some point" to withdraw the American troops now in South Korea.
The purchases are being finalized as Secretary of State Mike Pompeo claims that the USA can achieve the denuclearization of the Korean peninsula in just two and a half years.
"In coordination with our ROK (South Korean) partners, we will continue with our current military posture until we receive updated guidance from the Department of Defense and/or Indo-Pacific Command", the USFK said in a statement.
The sanctions on North Korea will not be lifted until progress toward denuclearization is verified.
It added that North Korea and the United States should take legal, institutional steps to guarantee it. In case of the USA and North Korea, it didn't happen as extensively as one would expect for North Korea is too isolated a country yet to engage in all-out diplomatic parleys.
However, the Pentagon has since denied that Mr Mattis was caught off guard.
Kim emphasized the need to take "practical measures" to implement all the agreements discussed during his summit with Trump and promised in the joint statement "at an early date". But, Koreans are anxious they could be affected by the summit discussion.
"We are working with China to a lesser extent".