He also expressed concern that the hermit state could be hit either by the coronavirus outbreak or a famine.
Mr Kim, 36, last appeared in state media on 12 April, triggering speculation that he was seriously ill.
But officials in South Korea later said such reports were not true.
There have also been suggestions that North Korea’s “supreme leader” may be staying at the sea resort of Wonsan to protect himself from possible exposure to coronavirus.
The secretive state shut its borders in late January due to the pandemic.
What did Mike Pompeo say?
“There is a real risk that there will be a famine, a food shortage, inside of North Korea too,” he added.
“We’re watching each of those things closely, as they have a real impact on our mission set, which to ultimately denuclearise North Korea,” the secretary of state said.
In the 1990s, a devastating famine is believed to have killed hundreds of thousands of North Koreans.
On Monday, President Donald Trump said he had a “very good idea” about Mr Kim’s condition, but added that “I can’t talk about it”.
“I just wish him well,” he added.
Mr Trump has met Mr Kim three times since 2018 – but the denuclearisation talks have stalled in recent months.
When did speculation start?
Kim Jong-un recently failed to appear for the celebration of his grandfather’s birthday on 15 April. This is one of the biggest events of the year, marking the birth of the nation’s founder.
Kim Jong-un has never missed this event – and it seemed very unlikely that he would simply choose not to turn up.
Inevitably, his absence prompted speculation and rumour, none of which is easy to substantiate.
Kim Jong-un last appeared in state media on 12 April “inspecting a pursuit assault plane group” in a handout that is undated. As ever, the images portrayed him as relaxed and at ease.
He chaired a key political meeting the day before, from state media despatches. But he has not been seen since.
The claim about Mr Kim Jong-un’s ill health first surfaced in a report for a website run by North Korean defectors earlier this month.
An anonymous source told the Daily NK that they understood he had been struggling with cardiovascular problems since last August “but it worsened after repeated visits to Mount Paektu”.
This led to a chain of reporting by international media on a single-sourced story.
News agencies then began to run with that claim, and it was all they had until some reports emerged that intelligence agencies in South Korea and the US were monitoring the claim.
But then came a more sensational headline in US media that the North Korean leader was in a critical condition after heart surgery.
However, a statement from the South Korean government, and sources at Chinese intelligence – speaking to the Reuters news agency – said this was not true.