The JCS said it called an emergency meeting of officials handling crisis management and operations just after noon Monday, after the North Korean media reported Kim's death.
The JCS said it has increased its monitoring activities along the border with the South Korea-U.S. Combined Forces Command. No unusual activity had been observed from the North, officials said.
Army Gen. Jung Seung-jo, chairman of the JCS, held an emergency meeting with U.S. Army Gen. James D. Thurman, the commander of the U.S. forces in Korea, to discuss monitoring activities. JCS officials said the generals agreed to bolster their sharing of information and intelligence in light of Kim's death.
The two also agreed not to raise the level of the Watchcon, an anti-North Korea surveillance alert, from its current level of three to two. Earlier Monday, South Korean officials had said Watchcon could be raised to a level that signifies a state of emergency with indications of a vital threat from North Korea.
"Since there is a great deal of shock within North Korea after the passing of its leader, the generals decided South Korea and the United States should not create an unnecessary sense of crisis," a South Korean military official said. "They said we should remain calm in our response while bolstering our defense posture, keeping in mind that Kim's death is something that took place in North Korea."
The South last raised its Watchcon to two immediately after the North shelled Yeonpyeong Island, just south of the Yellow Sea border on Nov. 23, killing two marines and two civilians.
Earlier Monday, officials said they may raise the "Defcon," a five-stage combat alert level, from its current level of four to three
"We're keeping close tabs on the demilitarized zone (DMZ), Joint Security Area (JSA) and Northern Limit Line (NLL) for possibilities of North Korean provocations," one JCS official said, referring to volatile border areas on land and in sea.
According to the JCS, the Master Control and Report Center (MCRC) at Osan Air Base, south of Seoul, which handles aerial monitoring and analysis, plans to ask for reinforcement by both South Korean and U.S. forces.