"I have every expectation we will be able to continue to do that", he said.
At the State Department on Monday, Pompeo said that the North Korean offer in Hanoi "simply didn't rise to the level that was acceptable given what they were asking for in exchange".
The US says sanctions will remain until Pyongyang destroys all nuclear sites.
"We don't know" what Kim will decide to do in the future and that any decision to resume such testing may "very much be his decision, and his decision alone", Biegun said.
Earlier, Ms Choe said USA diplomats had thrown away "a golden opportunity" during the summit when North Korea offered to dismantle its main Yongbyon nuclear complex.
Any launch would send the stuttering talks on denuclearisation into disarray, after the Hanoi summit broke up without agreement.
But Mr Pompeo said he is "hopeful" that discussions will continue.
"And I think in the president's judgment that time has arrived now", he said.
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.
She also said the North Korean leader will soon "clarify his position" on whether to continue talks or restart missile launches and nuclear tests.
"We have neither the intention to compromise with the U.S.in any form nor much less the desire or plan to conduct this kind of negotiation", Choe told reporters during a news conference in Pyongyang, according to the report.
The North Korean diplomat mixed these condemnations of Trump's top advisers with praise for the president himself, insisting the "chemistry" between Trump and Kim remains "mysteriously wonderful".
Earlier this week, Moon Chung-in, South Korean Special Adviser to the President for Foreign Affairs and National Security, told reporters that increasing sanctions wasn't an appropriate course of action if North Korea didn't denuclearize.
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"What is clear is that the US has thrown away a golden opportunity this time", she said. "I'm not sure why the United States came out with this different description". We never asked for the removal of sanctions in their entirety'.
What was the United States position?
USA 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.
"It was all about the sanctions".
In Washington this week, the US special representative for North Korea, Stephen Biegun, said the United States expected to be able to continue its close engagement, though he offered no specifics on when new talks might be held.
Instead, the USA president walked away from the table, and according to Vice Foreign Minister Choe, a member of the North Korean delegation complained about taking the long train ride to Hanoi only to return empty-handed. Their second meeting was in Hanoi in February.
He did, however, offer no specifics on when any new summit might be arranged.
North Korea claimed it had only requested a partial lifting of the sanctions. Pyongyang is aware that Donald Trump has boasted about his ability to get Mr Kim to stop firing missiles and testing nuclear missiles.
The activity at Sohae "doesn't necessarily really surprise me", Miller said.
After the two leaders failed to secure a deal in Hanoi, and with economic sanctions still in place, North Korea might be trying to hurry the U.S. president along and get him back to the table with a better deal.
Earlier Friday, the USA envoy for the DPRK Stephen Biegun urged the UN Security Council members to stay united in pressuring Kim Jong Un's government to give up its nuclear weapons following the failed talks between Trump and Kim in Hanoi last month. So the door to diplomacy is still open.
DPRK Vice Foreign Minister Im Chon Il on Thursday met with his Russian counterpart Igor Morgulov in Moscow amid speculation that DPRK leader Kim Jong Un could also pay his first trip to the neighboring country, reported Yonhap.