Trump and Kim met in Singapore on Tuesday for their countries' first-ever summit talks.
But Panetta, who previously worked for President Barack Obama, said there are a "huge number of issues" that will actually involve the denuclearization of North Korea, which owns an estimated 20 to 60 nuclear weapons with about 141 sites that can produce them.
North Korea has called for an end to the exercises as part of the United States' ending its "hostile policy" toward the country.
US President Donald Trump (2nd R) and North Korea's leader Kim Jong-un (2nd L) sign documents. "So, I think the rhetoric - I hated to do it, sometimes I felt foolish doing it - but we had no choice", he told Hannity.
In response to Trump's suggestion that an agreement to formally end the Korean War might take place soon, Nam said the Moon administration would not hasten the process and that there was "no timeframe" set for the process.
The summit was the first in history between a sitting U.S. President and a DPRK leader.
Following the unprecedented meeting in Singapore, Trump stunned observers when he said continuing the exercises routinely held between the US and South Korea would be "inappropriate" while Washington fleshes out a comprehensive deal with Pyongyang.
Few showers, storms possible Tuesday: Hot and humid rest of the week
Beyond Wednesday , rain chances become minimal as high temperatures creep back into the 90s for the second half of the week. The main threat will be focused across the southern half of the region from late afternoon through Wednesday morning.
If Mattis was aware that Trump was going to announce a suspension of military exercises, he apparently did not share that information with the South Koreans or with the military organization most directly affected: U.S. Forces Korea. Cory Gardner told reporters that they were assured joint military exercises would continue.
"We still look at Trump and the USA behavior with pessimism and we can not be optimistic about the USA behavior", Iranian Foreign Ministry spokesman Behram Qassemi said Monday, alluding to the USA president's dropping of JCPOA - the 2015 nuclear deal brokered by Barack Obama.
The president said he and Kim "got along very well" from "the beginning" of the summit.
And while denuking North Korea will certainly come, it will take time.
Trump also seemed annoyed that USA bomber aircraft make six-plus hour flights from the Pacific island of Guam to the Korean peninsula as part of its exercise routine.
A favoured US slogan in South Korea is "ready to fight tonight".
Exercises with South Korea cover a range of eventualities, including drills with nuclear-capable bombers.
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