Amid media reports that the Supreme Leader of North Korea, Kim Jong-un, is unwell and has undergone a surgery, US President Donald Trump on Tuesday wished him "well" but did not make any comment on the health of the reclusive leader.

"I can only say this: I wish him well," Trump told reporters at the White House, noting the "good relationship" he shares with the North Korean leader.

"I just hope he's doing fine," Trump said, adding he "may" reach out to Kim to see how he is doing. "If he is in the kind of condition that the reports say, that the news is saying, it's a very serious condition," he said.

Trump, however, did not mention if he had had any direct information about Kim's health conditions. His comments appeared as if he were basing them on news reports, admitting that he did not know if they were true or not.

This comes after shortly after US media, earlier yesterday, floated reports suggesting that North Korea's Supreme Leader had undergone a cardiovascular procedure and was now in "grave danger". South Korean government sources, however, downplayed these reports on Tuesday and said that Kim Jong-un was not "gravely ill"

According to a report by a Seoul-based publication, Daily NK, citing unidentified sources inside North Korea, it appears that Kim Jong-un is now recovering at a villa in the Mount Kumgang resort county of Hyangsan on the east coast after undergoing the cardiovascular procedure at a hospital on April 12.

Several South Korean media reports also corroborated that the speculations over Kim's supposed "grave health" conditions were fueled by the fact that the North Korean leader was absent from a key anniversary event earlier this month.

Pyongyang marked the birthday of its late founder, Kim's grandfather Kim Il Sung, on April 15 -- by far the most important date in its annual political calendar. The Supreme Leader was not seen in attendance.

Kim was last seen on April 11 in state media reports presiding over a political bureau meeting of the ruling Workers' Party, calling for "strict national countermeasures to thoroughly check the infiltration of the virus".

 US news media CNN reported that Kim was in "grave danger", citing unnamed US officials who reportedly told the publication that "the concerns about Kim's health are credible but the severity is hard to assess."

Trump and Kim have held two summits, beginning June 2018 in Singapore, which was the first-ever face-to-face meeting between leaders of the two countries.

The summit focused on security and denuclearization issues, but subsequent talks between the two leaders collapsed in Hanoi, Vietnam in 2019.