Thousands of American and South Korean troops were expected to participate in the exercise, the American portion of which is called Freedom Guardian.
"The Dokdo defence drill is a routine training conducted to prevent an invasion from external forces", said Choi Hyun-soo, a spokeswoman at Seoul's defence ministry. "Nothing has yet been decided, and it is still in discussion".
"This week, the South Korean and USA defence ministries will jointly announce the results of their discussions", the source added.
The conflict finished in an armistice agreement, not a peace treaty, leaving the two Koreas technically at war. The U.S. has long looked to China to use its influence with North Korea to bring it to negotiations, but the visit comes as ties between Beijing and Washington are being tested by a major trade dispute.
The president's announcement appeared to catch USA defense officials by surprise, and his comments ran counter to long-held American arguments that the exercises are critical for effective operations with allies and are defensive in nature.
Kim's third trip to China since March comes as Beijing tries to strengthen its role as a mediator between the U.S. and North Korea, where it claims compelling security and economic interests.
The U.S. has 6,800 nuclear warheads, 4,000 of which are stockpiled and 1,800 deployed, and 2,800 retired, according to a report from the Federation of American Scientists. This is Kim's second plane ride to China, according to NPR's Anthony Kuhn. He became the first North Korean leader to visit South Korea since 1953 in April.
Trump had vowed to prevent that happening and threatened North Korea with "fire and fury like the world has never seen".
The trip is Kim's third to China since March, when he made his inaugural foreign trip as leader.
The next exercise was scheduled for August with some 17,500 U.S. military personnel due to take part.
The announcement of the suspension of drills came as North Korean leader Kim Jong Un prepared to embark on a two-day visit to Beijing.
"There will be a point at which, when you're 20 per cent through (the denuclearisation process), that you (North Korea) can't go back", Mr Trump said. North Korea has said it will suspend nuclear and missile tests for seven months.
North Korea holds a massive artillery drill.
Ending the war games had been a long-standing demand by both North Korea and China.
"Secretary of State Pompeo has the will to push ahead with [negotiations] at a very fast speed", said Ms. Kang, who was personally briefed by Mr. Pompeo on the situation after the Singapore summit last week.
But it drew criticism as it lacked specifics, such as a timeline or a method for nuclear disarmament. They say the weapons are important to maintaining his family's leadership of the country. The two sides are discussing the possibility of sending a joint team to the upcoming Asian Games in Indonesia.
The president also called them "tremendously expensive".