After Kim missed the secretive country's most important national holiday, rumors of his demise have only grown, even as a new theory gains support -- social distancing against the coronavirus pandemic.
On Tuesday, Trump declined to answer questions about Kim, telling reporters at the White House, "I don't want to comment on it. I just wish him well."
The day prior, however, he suggested that Kim is alive and said that he had "a very good idea" of what his condition is.
Those cryptic remarks came days after Trump attacked CNN, which, citing an unnamed U.S. official, had reported that U.S. intelligence is assessing whether Kim is in "grave danger." At Thursday's briefing, he called that a "fake report" and said the network "used old documents" without explaining what he meant.
Given its invasive police state and severe isolation, North Korea is a difficult country for foreign intelligence agencies to penetrate and the inner workings of its ruling elite are tightly held -- making it difficult even for U.S. spies to know what's happening.
"Longtime watchers are just saying, 'Look, we don't know. No one knows,'" said Bruce Klingner, former deputy CIA chief for Korea and now a senior fellow at the Heritage Foundation.
A senior Defense Department official backed that up, telling ABC News on Sunday that anyone who says they know exactly what is happening with Kim does not know what they are talking about because the U.S. does not have clarity or sufficient intelligence penetration to know.
Kim was last seen at a communist party politburo meeting on April 11, leading the party's discussion on prevention of the novel coronavirus, according to photos published by state media. While he has been absent from state media for stretches of time before, including for nearly 20 days twice before this year, speculation rose because for the first time in his rule, he missed the commemorative events for the birthday of his grandfather and North Korea's founder Kim Il Sung on April 15 -- the country's most important holiday.
Beyond Trump, several senior South Korean officials have now also said that Kim is most likely alive and not in any kind of dire medical condition.
"So far, it is considered that there is no unusual movement in North Korea," South Korean Foreign Minister Kang Kyung-wha told parliament during a hearing on Tuesday.
Tune into ABC at 1 p.m. ET and ABC News Live at 4 p.m. ET every weekday for special coverage of the novel coronavirus with the full ABC News team, including the latest news, context and analysis.
The country's Minister of Unification Kim Yeon Chul added that North Korean state media reporting suggests Kim is handling his government's affairs as normal and suggested coronavirus prevention could be behind his absence instead.
Officially, North Korea's government has reported no cases of COVID-19, the disease caused by the coronavirus, and it has taken strict measures to shut its borders and restrict travel to stave off an outbreak.
"We don't have any information that there's an outbreak, but we are watching how they respond to it," a State Department official told ABC News last week. But several Korea watchers have said given North Korea's reliance on trade with China, from where the pandemic first spread, there are likely cases in the country.
Kim isn't the only one out of the public eye. His sister Kim Yo Jong, who has been discussed as a possible successor given the leadership's blood line, also missed the commemorative events for their grandfather's birthday on April 15.
Since the holiday, the core of North Korea's elites that often surround him have also been absent from public view, according to Thae Yong Ho, North Korea's former deputy ambassador to the United Kingdom who became the country's highest-ranking diplomat to defect in 2016.
That's fueled the theory that Kim and others are ensconced somewhere to keep them safe from a potential outbreak.
An official from South Korean President Moon Jae In's office told ABC News last week that Kim was out of the capital Pyongyang at a suburban retreat, while one of Moon's close advisers Moon Chung In said over the weekend that Kim is "alive and well" and in Wonsan.
Satellite images taken last Thursday by Planet Labs showed a train at the Leadership Railway Station in Wonsan, North Korea -- a city on the country's east coast where Kim has an expansive retreat complex. According to 38 North, a think tank that provides analysis on North Korea, the train which likely belongs to Kim, arrived some time after April 15, and "lend(s) weight to reports that Kim is staying at an elite area on the country's eastern coast."
Boats in the Wonsan area have also been notably active and "made unusual moves" in recent weeks, according to another specialty site NK News, "indicating his likely continued presence in the area," it reported Tuesday.
But Thae, now a member of parliament in the opposition conservative party, said Tuesday that the train could be a "disguise" because the government doesn't usually park the train where Kim actually is.
What's odd, Thae added, is that when North Korean Foreign Ministry staff outside of the country are asked about Kim's health, they normally fire back with aggressive denials, but this time, they are keeping quiet -- suggesting perhaps that even they don't know.
ABC News' Martha Raddatz contributed to this report.