The White House said Trump met with members of his national security team on Tuesday to discuss the growing threat from the North.
Suspicions that Pyongyang may have stolen intel from South Korea will do nothing to de-escalate tensions with the USA, which are already at a 50-year high following the North's rocket tests.
The information, that included wartime contingency plans drawn up by the US and South Korea, was from the country's defence ministry, according to Rhee Cheol-hee, a South Korean lawmaker.
The exercises came hours after a South Korean lawmaker claimed North Korean hackers had stolen a large cache of military documents from his country, "including a plan to assassinate North Korea's leader Kim Jong Un, and wartime contingency plans drawn up by the US and South Korea", reports the BBC.
The training was part of a programme of "extended deterrence" against North Korea, it added.
The statement by South Korea's Joint Chiefs of Staff also revealed that the two bombers carried out air-to-ground missile drills in waters off the east coast of South Korea, then flew over the South to waters between it and China to repeat the drill. Two days previous to that comment the president said on the social media platform that Pyongyang had violated agreements "before the ink was dry, making fools of USA negotiators. What I am going to tell you is that the [South Korea]-US alliance, that bilateral entity, is there to deal with those types of situation and safeguard against them".
China, North Korea's closest ally, has suggested that Pyongyang freeze its nuclear missile development while the United States stop conducting military exercises with the South that raise fears in North Korea.
North Korea has said Trump is "on a suicide mission" against it, with the country's foreign minister calling the US President "mentally deranged" in a speech at the United Nations last month.
Lee said that 235 gigabytes of military documents were taken, but the military has yet to identify 80 percent of the documents that were compromised.
North Korea watchers have been on alert for a possible provocation by the communist state in the days surrounding a key holiday Tuesday to mark the founding anniversary of its ruling party. Other stolen data included contingency plans for South Korean special forces and information on military facilities and power plants, he said.