President Donald Trump and North Korean leader Kim Jong Un walk after lunch at the Capella Hotel on Sentosa island in Singapore June 12, 2018.

North Korean leader Kim Jong Un said he wants to denuclearize the Korean peninsula during U.S. President Donald Trump's first term, as he agreed to hold a third summit with his South Korean counterpart this month in Pyongyang, Seoul officials said on Thursday.

Kim and South Korean President Moon Jae-in will meet in the North Korean capital on Sept. 18-20, during which they will discuss "practical measures" toward denuclearization, the South's national security adviser, Chung Eui-yong, told reporters a day after meeting Kim in Pyongyang.

Kim told the South Korean officials that his faith in Trump remains "unchanged" and he wanted to denuclearize and end long-standing hostile relations between North Korea and the United States during Trump's first term ending early 2021, Chung said.

Kim's remarks to South Korean officials mark the first time that the North Korean leader has offered a potential timeline for dismantling his country's nuclear weapons programme.

Kim "reaffirmed his determination to completely denuclearize" the peninsula, and expressed his willingness for close cooperation with South Korea and the United States in that regard, Chung said.

"Chairman Kim ... expressed frustration over the doubt shown by some parts of the international society about his will," Chung said.

"North Korea has been preemptively carrying out measures needed for denuclearization, and Kim said he would appreciate that such good faith is accepted with good faith."

South Korea's Moon sent Chung and other envoys to the North Korean capital on Wednesday to set the timing and agenda for the third inter-Korean summit this year, and to break the impasse in talks between Washington and Pyongyang over dismantling the North's nuclear program.

Kim and Trump held a watershed summit in June in Singapore, during which they said they would work toward complete denuclearization, establish "new" relations, and build "a lasting and stable peace regime" on the Korean peninsula.

But negotiations have since stalled over differences on how to implement the agreement, and amid signs of North Korea continuing with its weapons program.

Trump abruptly cancelled U.S. Secretary of State Mike Pompeo's planned visit to Pyongyang last month, citing a lack of progress in the talks.

During the Wednesday meeting, Kim emphasized that the initial steps North Korea has already taken — including dismantling its underground nuclear test site and a missile engine facility — means the country has ended nuclear and long-range missile testing "for good", Chung said.

"If reciprocation is shown for North Korea's preemptive measures that have already been completed, he strongly expressed his will that more active measures for North Korea's denuclearization can be taken."

Pyongyang has said the United States needs to reciprocate North Korea's goodwill gestures, which also include the return of U.S. war remains, by stating an official end to the 1950-53 Korean War.

But U.S. officials said they have already made concessions, with the suspension of U.S.-South Korean joint military exercises. They also worry that an end-of-war declaration could weaken North Korea's incentive for denuclearization, while raising questions about some 18,500 U.S. troops stationed in South Korea, a legacy of the Korean War.

Kim brushed off such worries on Wednesday, and said declaring an end of the Korean War has nothing to do with the U.S.-South Korean alliance or U.S. troops in South Korea, Chung said.