It marks the first time that three USA aircraft carriers have joined together at close range for a joint exercise with South Korea, officials said.
Instead the threat "pushes us to speed up the efforts to accomplish the great cause of completing the state nuclear force", the statement said, dismissing Trump's tour of the region as a "business trip".
The three carriers will be likely together in the drills around Monday, according to a South Korean military official, who didn't want to be named, citing office rules.
He directly addressed North Korea in a speech at South Korea's parliament on Wednesday, telling Kim Jong Un: "Do not underestimate us".
Trump is in Vietnam as part of a almost two-week tour of Asia, and North Korea's nuclear weapons program has been a major talking point with his Japanese, South Korean and Chinese counterparts.
The U.S. Navy's previous training operation involving three carriers was in the vicinity of Guam in the Western Pacific in 2007 during the Valiant Shield exercise.
The exercises will also involve 11 US Aegis ships and seven South Korean naval vessels, including two Aegis ships.
The U.S. carriers also participated in separate exercises with three Japanese destroyers on Sunday.
Seoul and Washington have recently carried out a series of drills to show off their military heft in response to the North's saber-rattling, such as its sixth and most powerful nuclear test on September 3 and a series of missile launches.
"The U.S., under my administration, is completely rebuilding its military, and they're spending hundreds of billions of dollars to the newest and finest military equipment anywhere in the world, being built right now". And do not try us.
During talks in Beijing Thursday, Trump urged Chinese leader Xi Jinping to work hard and act fast to help resolve the North Korean nuclear crisis, warning that "time is quickly running out".