Foreign doctors, possibly from China and France, performed the operation after Kim, 66, collapsed about Aug. 15, the newspapers Dong-a Ilbo and JoongAng Ilbo reported, citing unidentified government officials.
Kim's condition has improved and he is not suffering from slurred speech, a disability often associated with a stroke, the reports said.
If Kim were incapacitated, it could have serious implications for international negotiations on North Korea's nuclear disarmament. The talks recently hit a snag because of a dispute between North Korea and the U.S. over how to verify the North's nuclear programs, and a delay by Washington in its promised removal of North Korea from a list of nations that sponsor terrorism.
"I only hope that any situation happening in DPRK should not affect negatively what has been going on in terms of denuclearization process" U.N. Secretary-General Ban Ki-moon told a news conference on Thursday at the United Nations, using the initials of North Korea's official name.
"I'm also concerned deeply by DPRK's decision to go back to reassembling the nuclear facilities. They must commit to their agreement among the six-party talks for the early realization of the denuclearization process," said Ban, who was South Korea's foreign minister before taking the helm of the U.N. in January 2007.
Lee Cheol-woo, a South Korean ruling party lawmaker, said in a radio interview Thursday that Kim is "recovering fast," has "no problem speaking and communicating," and is "able to stand if assisted."
The lawmaker, a leader of the parliamentary intelligence committee briefed by the country's spy agency Wednesday, did not give further details.
However, South Korea's largest newspaper, Chosun Ilbo, said the stroke had left Kim with "partial paralysis." It quoted an unidentified senior government official as saying, "I understand that he is suffering inconvenience on the left part of his body."
South Korea's main spy agency declined to comment on the reports, only repeating a previous statement that Kim's condition had much improved from an unspecified circulatory problem. It also declined to say whether Kim received surgery.
On Wednesday night, the office of President Lee Myung-bak said it had received intelligence reports that Kim is recovering from a stroke and is still in control of his isolated country.
The North Korean leader was "not seen to be in a serious condition," presidential spokesman Lee Dong-kwan said in a statement after President Lee convened a security ministers' meeting to discuss the situation.
Defense Minister Lee Sang-hee told a parliamentary committee Thursday that the military does not plan to raise its defense alertness because North Korea's military has shown no unusual signs, a lawmaker's aide said.
"It appears that there is no leadership change" in North Korea, the defense chief told lawmakers during a closed-door briefing, according to Jun Eun-hye, an aide to ruling party lawmaker Yoo Seong-min.
Raising the defense alertness "could rather make the people uneasy and provoke North Korea," Lee was quoted as saying.
Still, the minister said the South is advancing its existing contingency plan to prepare for "any kind of situation whether it be limited or full-scale warfare," according to the lawmaker's aide.
Speculation about Kim's health intensified after he missed a parade Tuesday commemorating the communist state's founding 60 years ago. That followed weeks of absence from public view and rumors that foreign doctors had been called in to treat him.
North Korea tried to dispel the rumors about his health.
"There are no problems," No. 2 leader Kim Yong Nam told Japan's Kyodo News agency.
Despite the willingness of North Korean officials to speak through a foreign news agency, their own state media remained quiet about Kim's condition.
South Korea's Yonhap news agency, citing lawmakers briefed by the spy agency, said Kim suffered a cerebral hemorrhage, but was conscious and "is able to control the situation." The report did not say when he suffered the stroke.
The spy agency also reported to lawmakers that Kim is in a "recoverable and manageable condition," and that North Korea is not in a "power vacuum," Yonhap said.
"If he had surgery, it means it's serious," said Kim Jong-sung, a neurology professor at Seoul's Asan Medical Center.
A cerebral hemorrhage can result in death, paralysis, difficulty in speaking and other disabilities, although if it is minor, recovery is possible without long-term affects. Surgery is generally only considered in the most serious cases, he said.
Kim, who has been rumored to be in ill health for years, took over North Korea after the death of his father in 1994.