Pompeo met with the North Korean leader in Pyongyang on Sunday to rekindle stalled denuclearization talks following a landmark summit between Kim and US President Donald Trump in Singapore, AFP reported.
"Chairman Kim said he's ready to allow them to come in" and see the dismantled Punggye-ri nuclear test site, Pompeo said.
North Korea took apart the Punggye-ri facility in the country's northeast in May but has yet to allow international observers into the site to verify its claims.
The facility, buried inside a mountain near the border with China, was the staging ground for all six of the North's nuclear tests.
The inspectors will be allowed in as soon as the two sides agree on "logistics", Pompeo told reporters in Seoul before leaving for Beijing on a whirlwind diplomatic trip.
Pompeo, however, did not comment on possible corresponding measures to be taken by Washington.
Denuclearization of North Korea is "a long process", Pompeo said, adding: "We made significant progress."
The top US diplomat also said the two countries were "pretty close" to setting a date and location for the second summit between Kim and Trump.
The visit was Pompeo's fourth to North Korea.
Trump met Kim in Singapore in June for the first-ever summit between the two countries, resulting in what critics say was only a vague commitment by the North Korean leader toward denuclearization.
Kim "expressed his gratitude to President Trump for making sincere effort" to implement the Singapore agreement, according to a report on the Pompeo visit by North Korea's state news agency KCNA.
"Kim Jong-un appreciated the positively developing situation on the Korean peninsula... (and) explained in detail the proposals for solving the denuclearization issue," KCNA said.