President Trump on Wednesday teed up another summit with North Korean dictator Kim Jong-un, claiming major progress toward denuclearization and lining up countries to help usher in an economic renaissance in the reclusive communist state.

The offer of economic prosperity that would go far beyond merely lifting sanctions on Pyongyang was the crux of Mr. Trump’s pitch for Mr. Kim to give up nuclear weapons when they met at a summit June 12 in Singapore.

Mr. Trump said Wednesday that he would meet again with Mr. Kim in the “very near future” and promised an announcement soon of the time and place for the summit.

The leaders could meet as soon as next month but more likely after that, Secretary of State Mike Pompeo said on “CBS This Morning.”

Mr. Trump has been increasingly upbeat about diplomacy with North Korea, with the denuclearization deal back on track after some rough months following the Singapore summit.

Despite some hot rhetoric from Pyongyang during that rough patch, the communist country has not conducted a nuclear test in more than a year nor a missile test in nearly a year.

“I think we are doing much better than anyone would know,” Mr. Trump said at a meeting with Japanese Prime Minister Shinzo Abe on the sidelines of the United Nations General Assembly meeting in New York.

Mr. Trump said Mr. Abe’s country stands ready to help develop the economy in North Korea, the most undemocratic country in the world where, despite having nuclear weapons, most people live in extreme poverty.

“Chairman Kim sees a great future for North Korea, and I firmly believe and I know Japan wants to help him toward that future economically. And we all want to help,” Mr. Trump said.

He said South Korean President Moon Jae-in, with whom Mr. Trump met Tuesday, also was eager to help.

Mr. Trump said he received “an extraordinary letter” from Mr. Kim that promised more progress in ending the 65-year-old military standoff between the U.S. and North Korea.

“He wants to see things happen for North Korea that are great that can make it an economic power, and they really have the potential to be that,” he said.

Mr. Trump earlier told the U.N. Security Council that many positive developments on North Korea were happening behind the scenes “away from the media,” and he promised “very good news” from North Korea in the coming months.

He also called on the U.N. Security Council to strictly enforce its sanctions on North Korea and noted that some countries were violating the sanctions. He was likely referring to China and Russia.

“I think we will make a deal,” he said.

“But unfortunately to ensure this progress continues, we must enforce existing U.N. Security Council resolutions until denuclearization occurs,” Mr. Trump said.

Mr. Pompeo said he will travel to Pyongyang next month to prepare for the second Trump-Kim summit.

He announced the tip on Twitter after what he called a “very positive” meeting with North Korean Foreign Minister Ri Yong-ho, also on the sidelines of the U.N. gathering.

“Much work remains, but we will continue to move forward,” he said after the meeting.

The State Department said Mr. Pompeo accepted an invitation from Mr. Kim to travel to Pyongyang next month for what will be his fourth visit this year.

Referring to North Korea by the acronym of its official name, the Democratic People’s Republic of Korea, the State Department said the aim would be “to make further progress on the implementation of the commitments from the U.S.-DPRK Singapore summit, including the final, fully verified denuclearization of the DPRK, and to prepare for a second summit between President Trump and Chairman Kim.”

• Gabriella Muñoz contributed to this article, which is based in part on wire service reports.

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