The rumors illustrate North Korea’s outsize dependence on a single person for its political stability. The answer to the question of his health holds significant implications for the future of high-stakes nuclear negotiations with the U.S. and the stability of a country on China’s border.

“These persistent rumors suggest that there are concerns that his health could impact his ability to lead the country,” said Jean Lee, a public-policy fellow at the Washington-based Woodrow Wilson Center. “And those concerns are always worth paying attention to with a country like North Korea where the system has been built around the rule of one family.”

But the flow of information from the secretive state is so poor, especially when it comes to the health of its leaders, that all longtime experts could do was hypothesize, advise caution and wait.

“This is a typical black-swan scenario,” said Go Myong-hyun, a research fellow at the Asan Institute, a Seoul-based think tank. “Kim Jong Un’s death is a low-probability event. But his death would be a game changer.”

Mr. Kim—as well as his father and grandfather, who preceded him in ruling the cloistered state—has a history of unexplained absences. He disappeared for more than a month in 2014 before ultimately reappearing with a cane. He was also out of the public eye for several weeks in 2012 and 2013.

Mr. Kim’s father,

Kim Jong Il,

disappeared in 2008, triggering rumors of ill health. South Korean officials later learned that the elder Mr. Kim had suffered a stroke, accelerating preparations within the regime for a succession plan. When he did die, in 2011, the world didn’t learn of it for two days.

In November 1986, South Korea’s Defense Ministry told reporters that

Kim Il Sung,

North Korea’s ruler at the time, had been killed, citing broadcasts from loudspeakers at North Korea’s front-line military units on the inter-Korean border.

The North’s state media published a report a few days later saying he had greeted Mongolian guests at Pyongyang’s airport.

The Asan Institute’s Mr. Go said the most conspicuous aspect of the current rumor cycle was North Korea’s silence.

“It is noteworthy that North Korea has neither confirmed nor released a statement laughing at the rumors,” he said.

Mr. Kim was last shown in the country’s state media on April 12, with images of him leading discussions on the new coronavirus pandemic and visiting an air force base. He missed the April 15 public ceremonies marking the birthday of Kim Il Sung, North Korea’s founder and Mr. Kim’s grandfather.

The past week has offered rich material for speculation about Mr. Kim’s health. On April 20, Daily NK, a Seoul-based outlet claiming to have sources inside North Korea, reported Mr. Kim was recovering at one of his countryside villas after having cardiovascular surgery.

Later in the week, an article in the Japanese magazine Shukan Gendai cited an unnamed Chinese medical official saying Mr. Kim appeared to have had a heart attack and was given emergency surgery by North Korean doctors to insert arterial stents. The operation took longer than it should have, and Mr. Kim entered a vegetative state, the magazine said, adding that Chinese doctors flew in but couldn’t get there in time to help with the operation.

On Saturday, Reuters reported China had dispatched medical experts to North Korea, possibly to help care for Mr. Kim.

Separately, analysts at 38 North looked at satellite data and determined that a train probably belonging to Mr. Kim was at the station near the leader’s compound at Wonsan, on the country’s east coast. In an article on Saturday, the North Korea-focused website said it meant Mr. Kim could be at the compound—hours away by road from where he was reported to have been recovering from surgery.

South Korean officials said throughout the week that they hadn’t picked up military movements or other signs suggesting a sudden change inside North Korea.

A South Korean lawmaker on the intelligence committee hadn’t seen signs suggesting special developments inside North Korea after a briefing by officials with South Korean‘s spy agency, his aide said.

U.S. officials have said they are looking into reports that Mr. Kim may have been incapacitated or worse.

Robert Carlin, a former U.S. intelligence official, said there is no credible evidence of anything being wrong with Mr. Kim, and that other reasons might explain his absence from the public scene, even from the April 15 commemoration.

“He could be trying to get out from under the shadow of his father and grandfather,” Mr. Carlin said. “So far, this is not a long absence, but if this goes on for a few more weeks, that would begin to raise some questions.”

President Trump, asked about Mr. Kim’s health during a White House coronavirus briefing on Thursday, said he believed a report on Mr. Kim was erroneous: “I hear the report was an incorrect report. I hope it was an incorrect report.”

A sudden change at the top of North Korea would alarm national-security officials in the U.S. and its allies, experts said, amid concern about the country’s nuclear arsenal and armed clashes spilling over the border into South Korea. China would worry about instability and a flood of refugees.

Mr. Kim’s younger sister,

Kim Yo Jong,

believed to be in her early 30s, has been cited as a potential successor. She is a close adviser to Mr. Kim and has blood ties, said Choi Kang, the vice president of the Asan Institute. She has attended Mr. Kim’s summits with President Trump and South Korean President

Moon Jae-in.

She also visited South Korea during the 2018 Winter Olympics.

Mr. Kim’s wife, Ri Sol Ju, and half-uncle, Kim Pyong Il, have also been mentioned as possible successors.

Mintaro Oba, a former U.S. State Department official, said the theory that Kim Jong Un has been dealing with medical issues is plausible. Mr. Kim, believed to be 36 years old, has long attracted concerns about his longevity because of his smoking and drinking habits and his apparent obesity. But Mr. Oba said the level of unfounded speculation was running unusually hot.

“His disappearance from public events matters, but it’s certainly not unprecedented, and we just don’t have enough information at this point to determine why,” he said. “Could he be dead or gravely ill?  Yes. Could it be something much more minor? Also yes.”

—Alastair Gale contributed to this article.

Write to Andrew Jeong at andrew.jeong@wsj.com