But Bolton also said the U.S. has a plan for Kim to denuclearise within a year, something Secretary of State Mike Pompeo will discuss when he meets with North Korean officials this week.
U.S. secretary of state Mike Pompeo will make another trip to North Korea this week.
Sung Kim is a former US envoy for North Korea and led policy negotiations with Pyongyang in advance of the summit.
In a sign the two sides are stepping up diplomacy, the USA ambassador to the Philippines, Sung Kim, led a delegation to the tense border area on Sunday in the highest-level meeting with the North since the June 12 summit.
U.S. officials say they are clear-eyed about the North's past behaviour in negotiations.
Kim agreed at the June 12 summit to "work toward denuclearisation of the Korean Peninsula", but the joint statement released after the meeting gave no details on how or when Pyongyang might forsake its nuclear and missile programs.
"None of that activity is in violation of any agreements made at the Singapore summit between US President Donald Trump and North Korean leader Kim Jong-un", Vipin Narang, an MIT political science professor and specialist on nuclear proliferation, told BBC News.
The Pentagon declined to address intelligence reports on North Korea's nuclear capability when asked on Monday, saying only that the USA military remains postured to "deal with any and all threats on the peninsula" with the goal of allowing "diplomats the space and the time to hopefully make progress coming out of the summit".
North Korea and Washington have yet to negotiate the terms under which the North would relinquish its weapons, so Pyongyang can be expected to seek leverage in those discussions.
Instead, there are indications that instead of stopping its nuclear and missile programs, Pyongyang is continuing to bulk them up.More news: US, Canada trade war?
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Samantha Vinograd, President Barack Obama's former Director for Iraq on the National Security Council warned that North Korea is deceiving President Donald Trump and he doesn't seem too anxious about it.
The meeting comes amid growing scrutiny of Pyongyang's commitment to denuclearization and as joint efforts to repatriate the remains of US troops from North Korea are taking longer than many anticipated.
"I am sure that Secretary of State Mike Pompeo will be discussing this with the North Koreans in the near future about really how to dismantle all of their [weapons of mass destruction] and ballistic missile programs in a year", Bolton said. It's to North Korea's advantage to see these programmes dismantled very quickly, because then the elimination of sanctions, aid by South Korea and Japan and others can all begin to flow.
Some experts have said a one-year program would be hard, with Stanford University's Siegfried Hecker, a leading expert on the North's nuclear program, proposing a 10-year roadmap instead.
Separately, he told "Fox News Sunday" that "we are using the full range of our capabilities to understand what North Korea is doing".
Given North Korea's long history of deception, such reports are likely to raise more questions about exactly what was achieved at Singapore.
USA officials have been vague or dismissive when asked whether Pyongyang has provided a full and detailed accounting of its nuclear program and arsenal - a requirement if monitors are to reliably monitor its destruction.
"Now we're saving a lot of money", Trump said of the cancellation of large-scale military drills that involve flights of U.S. bombers from the Pacific U.S. territory of Guam.
"I do not believe it would be possible to verify full dismantlement within a year, nor have I yet seen evidence of a firm DPRK decision to undertake full dismantlement". Even so, the president quickly hailed the meeting as a major breakthrough, going so far as to tweet that North Korea "is no longer a Nuclear Threat".
He said the one-year program the U.S.is proposing would cover all the North's chemical and biological weapons, nuclear programs and ballistic missiles.