Pompeo was in Pyongyang earlier this month to flesh out a deal reached by US President Donald Trump and North Korean leader Kim Jong-un at their historic summit in Singapore in June.
According to Rep. Roh Hoe-chan, floor leader of the minor opposition Justice Party, the top US diplomat proposed that the two sides discuss three things: a full declaration of North Korea's nuclear arsenal, a timeline for dismantling the nuclear program, and an unfulfilled promise made by Kim at the summit.
"North Korea gave no response," Roh told Korean reporters in Washington, saying he was informed by acting US Under Secretary for Political Affairs Stephen Mull.
"I am guessing North Korea responded that they should first take a confidence-building step together, for example by declaring an end to the (1950-53 Korean) war," he said.
The unfulfilled promise referred to by Roh was likely what Trump has said was a commitment made by Kim during the summit to destroy a missile engine testing site in the North.
The Korean War, in which the US fought alongside the South, ended in an armistice, not a peace treaty, leaving the two sides technically at war.
The first-ever meeting between a US and North Korean leader yielded a commitment by Kim to work towards the "complete denuclearization" of the Korean Peninsula in exchange for security guarantees from the U.S.
Pompeo reported after his trip that the talks were "productive," but acknowledged that a lot of work remained.
Trump and Kim exchanged threats and personal insults last year as North Korea tested its sixth nuclear weapon and ballistic missiles capable of reaching the US.
Trump declared soon after the summit that the nuclear threat was now over.
But various news reports since have cited US intelligence agencies as detecting a continued expansion of North Korea's nuclear and missile facilities.
Trump has also backed away from his previous calls for swift denuclearization, saying recently that there is "no rush" and "no time limit."
Roh was in Washington as part of a bipartisan National Assembly delegation seeking to advance South Korea's interests in denuclearization and trade with the US government and Congress. (Yonhap)