South Korean and Chinese officials have cast doubt on reports after media outlets said he had undergone a cardiovascular procedure and was in "grave danger".
Daily NK, a Seoul-based speciality website, reported citing a source in North Korea that Mr Kim was recovering after undergoing the procedure on 12 April.
The North Korean leader is believed to be about 36.
CNN cited a US official with direct knowledge as saying Washington was "monitoring intelligence" that the leader was in grave danger after surgery.
Bloomberg quoted an unnamed US official as saying the White House was told that he took a turn for the worse after the surgery.
However, two South Korean government officials rejected the CNN report without elaborating on whether Mr Kim had undergone surgery.
The presidential Blue House said there were no unusual signs coming from the reclusive, nuclear-capable state.
Mr Kim is the unquestioned leader of North Korea and the sole commander of its nuclear arsenal.
He has no clear successor and any instability in the country could be a major international risk.
The state KCNA news agency gave no indication of the whereabouts of the leader in routine dispatches on Tuesday, but said he had sent birthday gifts to prominent citizens.
An official at the Chinese Communist Party's International Liaison Department dealing with North Korea told Reuters the source did not believe Mr Kim was critically ill. China is North Korea's only major ally.
Chinese foreign ministry spokesman Geng Shuang said Beijing was aware of reports about Mr Kim's health, but did not comment on whether it has any information about the situation.
Daily NK earlier said the leader's health had deteriorated since August due to heavy smoking, obesity and overwork.
It said he was now receiving treatment at a villa in the Mount Myohyang resort north of the capital Pyongyang.
Mr Kim's potential health issues could fuel uncertainty over the future of the reclusive state's dynastic rule and stalled denuclearisation talks with the United States, issues in which he wields absolute authority.
With no details known about his young children, analysts say his sister and loyalists could form a regency until a successor is old enough to take over.
Speculation about the reclusive leader's health first arose following his absence from the anniversary of the birthday of its founding father and Kim's grandfather, Kim Il-sung, on 15 April.
Reporting from inside North Korea is notoriously difficult, especially on matters concerning the country's leadership, given tight controls on information. There have been false and conflicting reports in the past on matters related to its leaders.
Mr Kim is a third-generation hereditary leader who rules North Korea with an iron-fist, taking over the titles of head of state and commander in chief of the military since late 2011.
