North Korea is considering suspension of denuclearization talks with the United States, Deputy Foreign Minister Choe Son Hui told reporters on Friday.
Pompeo told reporters on Friday at the White House that North Korea's name-calling had not done anything to harm talks with Kim.
He did, however, offer no specifics on when any new summit might be arranged.
South Korea's presidential Blue House said in a statement: "We can not judge the current situation based exclusively on Vice Minister Choe Son Hui's statements".
"Following that, we continued to have very professional conversations, where we tried our best to work together and represent our respective sides. We'll look at ramping those sanctions up in fact", Bolton said.
He downplayed recent satellite images that analysts say show activity at North Korean missile sites and urged against making any "snap judgment" on the significance of the images that appear to show that North Korea has begun rebuilding a portion of the Sohae facility previously used to test long-range missile engines.
North Korea threatened on Friday to suspend negotiations with the US and resume nuclear and missile testing, BBC News reported.
Pompeo gave no sign of USA willingness to soften its stance in demanding that North Korea give up its nuclear weapons.More news: Mumbai bridge collapse: What has happened so far
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U.S. national security adviser John Bolton, who has argued for a tough approach to the North, said last week that Trump was open to more talks but warned of tougher sanctions if Kim did not denuclearise.
"That's Chairman Kim's word", Mr Pompeo said.
Pyongyang may freeze talks with the U.S. and renew missile and nuclear tests after senior USA officials torpedoed President Trump's talks with Kim Jong-un in Vietnam, a top North Korean diplomat said. Talks between Mr Trump and Kim - the second time the two leaders have met - collapsed as a result the latter's demands over worldwide sanctions imposed in response to nuclear weapons programme, which it pursued for years in defiance of UN Security Council resolutions.
Bolton said Friday he has spoken with his South Korean counterpart about the North Korea allegations. Instead she blamed Mike Pompeo and National Security Adviser John Bolton for hardening the United States position.
Pompeo said that in her remarks, Choe had left open the possibility that negotiations would continue.
The North´s warning came amid concerns over its satellite rocket launch site, where some rebuilding activity has been observed in recent weeks, triggering worldwide alarm that Pyongyang might be preparing a long-range missile or space launch. Hanoi was supposed to be the part of the same old story where the US accepted a meager offer from North Korea and declared diplomatic victory, only to watch the North Koreans take back everything they "gave away" a few months later and demand fresh concessions to refrain from further provocations.
Commercial satellite images, released by US -based think tanks, suggested the North may be rebuilding its Dongchang-ri long-range rocket test site close to the border with China. "I have every expectation we will be able to continue to do that".
South Korea, which has an ambitious agenda of engagement with North Korea that is dependent on Pyongyang and Washington resolving at least some of their differences, said it was too early to tell what Choe's comments might mean.
"As long as there's no testing", said Mr Trump, "I'm in no rush".