If North Korea uses the drills this week as a reason to launch missiles around Guam or elsewhere, it could set off a new cycle of escalation.
Just last week, Pyongyang said it had finalized a plan to fire four missiles toward the USA territory of Guam.
Past drills are believed to have included simulated "decapitation strikes", which see soldiers carry out trial operations to kill Kim Jong Un and his top generals.
This is expected to be a watershed in the North Korean nuclear standoff.
This article has been amended to include comments made by Moon Jae-in to a delegation from the U.S. Senate.
North Korea also warned the USA on Sunday of "adding fuel to fire" by proceeding with the planned war games, calling it a "reckless" move that could trigger the "uncontrollable phase of a nuclear war". It comes on the heels of North Korea testing two intercontinental ballistic missiles in July, and an announcement that it was examining a plan to possibly bracket Guam with missiles - a plan that is on hold.
"It will take resolute steps the moment even a slight sign of the preventive war is spotted".
It did not provide any details on what it meant by "preventive war".
North Korea has warned the United States will be "pouring gasolene on the fire" by conducting a war game with its long-time southern foe.
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Moon said North Korea had not stopped its efforts to advance its nuclear and missile technologies, even as the worldwide community strengthened pressure and sanctions on Pyongyang.
The Rodong Sinmun, an organ of the ruling Workers' Party of Korea (WPK) attacked the drills on Sunday, in a commentary headlined "Foolish behavior to hasten self-destruction".
China, North Korea's closest ally, had previously called for this year's military exercise to be dropped amid fears it would provoke the North even more as tensions have spiked. "The problem is that if the Second Korean War occurs, it will only be turned into a nuclear war".
US Pacific Command chief Admiral Harry Harris arrived in the South on Sunday to inspect the exercises and discuss growing North Korean nuclear and missile threats.
DPRK stands for the Democratic People's Republic of Korea, North Korea's official name.
Moon said the current situation is more grave than ever before. "The people can be assured that there will be no war", Moon said on August 17.
With tensions high on the Korean peninsula, China - Pyongyang's main ally and trading partner - has urged the United States and South Korea to scrap the drills, as has Russian Federation.
South Korea called the North an enemy for the first time in 2010 under the conservative Lee Myung-bak government, and has kept that expression since then. Pyongyang has denounced the drills as a rehearsal for a northern invasion.
"If the U.S. president did move unilaterally in a circumstance like that, against the [wishes] of a South Korean president, the U.S. -ROK defense alliance would be over in a heartbeat", he said.