U.S. Secretary of State Mike Pompeo on Sunday said he had "good, productive conversations" with North Korean leader Kim Jong-un during his meeting in Pyongyang, according to the Yonhap news agency.

Pompeo then traveled to Seoul to meet South Korean President Moon Jae-in and Foreign Minister Kang Kyung-wha.

"As President [Donald] Trump said, there are many steps along the way and we took one of them today. It was another step forward," said Pompeo in his meeting with South Korean leader.

"So, this is, I think, a good outcome for all of us," Pompeo said.

Pompeo thanked South Korea for its role in the negotiations with the Pyongyang administration about nuclear weapons.

Moon said the world's eyes "are on the outcome" of their meeting.

"Today is a very important day for the U.S. as well as South and North Korea," Moon said. "I hope Secretary Pompeo's North Korea visit and the upcoming second U.S.-North Korea summit will create good momentum for a breakthrough in the denuclearization and irreversible progress in the peace process on the Korean Peninsula."

In a tweet, Pompeo later said: "We continue to make progress on agreements made at Singapore Summit. Thanks for hosting me and my team @StateDept."

The U.S. President Donald Trump also tweeted about Pompeo's visit to Koreas.

"@SecPompeo had a good meeting with Chairman Kim today in Pyongyang. Progress made on Singapore Summit Agreements! I look forward to seeing Chairman Kim again, in the near future," Trump sad.

In June, Trump offered a brighter economic future to North Korea and the lifting of sanctions, on the condition that Kim would verifiably denuclearize his country.

Dialogue between Washington and Pyongyang subsequently broke down over a lack of progress, and apparent reluctance on the part of the U.S. to agree to simultaneous measures such as officially ending the Korean War in advance of the North's denuclearization.