"The idea that we're going to relax the sanctions just on North Korea's say-so, I think, is something that just isn't under consideration", Bolton said.
The statement cited US officials who are "going against President Trump" and his intent to advance relations between the two nations, saying that the president has expressed "gratitude to our goodwill measures" implemented as part of the agreement.
The rival Koreas plan to hold high-level talks on Monday to prepare for a third summit between their leaders, as Pyongyang called on the United States to reciprocate its "goodwill measures" by easing sanctions and stopping demands that the North denuclearize first.
He continued to note that the US has no current plans to relax sanctions on North Korea until there is evidence the country is taking steps to denuclearize, saying the Trump administration would continue its campaign of "maximum pressure".
North Korea appears to continue dismantling a rocket launching site, a USA think tank said, based on satellite images.
But the North Korea statement separates Mr. Trump from "high-level officials", thus allowing the the country to rail against the USA, while leaving the door open for the president to continue his dialogue with leader Kim Jong Un.
The White House and State Department did not immediately return requests for comment.
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Pyongyang insisted that the US should reciprocate to the North's suspension of missile launches and nuclear tests and other goodwill gestures such as the return of remains of American troops killed in the Korean War. Those followed contentious remarks last week by US Secretary of State Mike Pompeo and North Korean Foreign Minister Ri Yong-ho on the sidelines of a regional forum in Singapore.
Reporting on preliminary results from a 10-month investigation, the Korea Customs Service said Friday it is seeking prosecutions of three local companies and their executives for smuggling or forging documents to say North Korean mineral resources came from Russian Federation.
But, following the meeting, USA intelligence officials said they believed North Korea was "deceiving" the US, saying the regime was bolstering production for nuclear weapons at "multiple secret sites" in recent months. They said that when a session with Pompeo reached an impasse last month, North Korean officials asked if he would like to step outside to call the president, who declared after the June summit that North Korea was "no longer a nuclear threat".
On Wednesday, U.S. Ambassador to the United Nations Nikki Haley said the United States was "not willing to wait for too long" for North Korea to take steps toward denuclearising.
The concessions the USA would offer in exchange - beyond sanctions relief or removing North Korea from the list of state sponsors of terrorism - remain unknown.
"In the current situation, friendly countries should develop their relations and cooperation in (the) worldwide community", he said, adding Iran and North Korea have "always had close views" on many issues.
The summit held in Singapore marked the first time Kim Jong-un had met a western leader.