The on-again, off-again summit seems to be on again.

Amid renewed hope Sunday that a summit next month between President Trump and North Korean leader Kim Jong-un will actually happen, South Korea’s president said that Mr. Kim committed over the weekend to a denuclearization of the peninsula and said he wants to work toward peace.

A U.S. delegation arrived in North Korea on Sunday as preparations for the talks continue. Mr. Trump announced last week he was ending them over North Korean hostilities but now says negotiations have picked back up and he spoke Sunday as if the summit was a “go.”

“Our United States team has arrived in North Korea to make arrangements for the Summit between Kim Jong-un and myself. I truly believe North Korea has brilliant potential and will be a great economic and financial Nation one day. Kim Jong-un agrees with me on this. It will happen!” Mr. Trump tweeted Sunday afternoon.

U.S. lawmakers said that while renewed discussions about the Trump-Kim summit are a positive step, the U.S. can’t give away too much and should be wary of any supposed commitments coming from Mr. Kim.

But South Korean President Moon Jae-in said Sunday that the reclusive North Korean leader is indeed committed to a full denuclearization of the peninsula, after the two leaders gathered in an impromptu meeting Saturday in hopes of jump-starting diplomatic talks once more.

Mr. Moon told reporters Sunday that Mr. Kim at a meeting at the Demilitarized Zone that separates the two Koreas “again made clear his commitment to a complete denuclearization of the Korean Peninsula.” According to Mr. Moon, the North Korean leader said he’s willing to cooperate to end confrontation and work toward peace for the sake of a successful North Korea-U.S. summit.

Mr. Moon said he told Mr. Kim in return that Mr. Trump has a “firm resolve” to end hostile relations with North Korea and initiate economic cooperation if Pyongyang implements “complete denuclearization.”

According to the State Department, the U.S. team met with North Korean officials at the same DMZ site, the village of Panmunjom. The U.S. delegation reportedly included Sung Kim, U.S. ambassador to the Philippines and a former U.S. special representative for North Korea policy.

“We’re looking at June 12 in Singapore. That hasn’t changed,” Mr. Trump told reporters earlier the weekend.

Mr. Trump called off the summit Thursday following threats of nuclear war from North Korea, only to give fresh hope Friday that negotiations to stage the summit were in the works.

“I think people want to see if we can get the meeting and get something done,” Mr. Trump said. “We get that done and we can be successful in the denuclearization of the Korean peninsula, that would be a great thing for North Korea, it would be a great thing for South Korea, it would be great for Japan, it would be great for the world, it would be great for the United States, it would be great for China.”

But Mr. Moon also made clear that any steps to dial back Mr. Kim’s nuclear capabilities could come with strings attached.

“What Kim is unclear about is that he has concerns about whether his country can surely trust the United States over its promise to end hostile relations (with North Korea) and provide a security guarantee if they do denuclearization,” Mr. Moon said.

The North has previously used the term “denuclearization” to demand the United States pull out its 28,500 troops in South Korea and withdraw its so-called “nuclear umbrella” security commitment to South Korea and Japan. The North hasn’t specified those demands during Mr. Kim’s sudden outreach to Seoul and Washington in recent months.

Mr. Moon has insisted Mr. Kim can be persuaded to abandon his nuclear facilities, materials and bombs in a verifiable and irreversible way in exchange for credible security and economic guarantees. Mr. Moon said Sunday that the North’s disarmament could be still be a difficult process even if Pyongyang, Washington and Seoul don’t differ over what “complete denuclearization” of the peninsula means.

Members of Congress said the devil, as is always the case with North Korea, will be in the details.

“We’ve got the South Korean president, I believe, providing a great benefit in this discussion, unless we are not all communicating in the same way,” said Sen. Roy Blunt, Missouri Republican, on “Fox News Sunday.”

“You know, what does denuclearization mean? What do they expect to see happen? And I think these talks can produce results. But we need results and they need to be very visible before the United States creates any benefits,” he said.

Sen. Marco Rubio, Florida Republican, said he’s skeptical that Mr. Kim will agree to completely give up his country’s nuclear capabilities, saying the leader has an almost emotional attachment to them and that they make him feel prestigious and powerful.

“Ultimately, I remain convinced that he does not want to denuclearize — in fact, he will not denuclearize. But he wants to give off this perception that he’s this open leader, that he’s peaceful, that he’s reasonable,” Mr. Rubio said on ABC’s “This Week.”

Sen. Chris Coons, Delaware Democrat, said even offering North Korea a summit with the U.S. president is a significant concession, while also saying it’s important that North Korea has freed several Americans the country was holding as a confidence-building measure.

“But let’s be clear — twice before North Korea under Kim Jong-un’s father committed to full denuclearization in six party talks in 2005, in 2007 … only to then change their tune,” Mr. Coons said on “Fox News Sunday.”

“So we’re going to have to be very focused, very engaged,” Mr. Coons said. “I am hopeful that Secretary [of State Mike] Pompeo will brief and engage Congress. And I’m hopeful that we’ll work together to make sure that this summit is a success.”

Even Mr. Trump’s critics have credited the president for his negotiating skills in his recent dealings with Mr. Kim, and calling the summit a positive step.

“In some ways, Kim Jong-un may have met his match here with our very unconventional president,” said former Director of National Intelligence James Clapper on CNN’s “State of the Union.”

Mr. Clapper, who’s publicly butted heads with Mr. Trump over issues related to Russian interference in the 2016 U.S. election and special counsel Robert Mueller’s probe, said a summit would be good regardless of whether there is any sort of agreement.

“Having gone this far, there is value in meeting and greeting, gripping and grinning, and just establishing a rapport,” he said. “Yes — I think it would be important to have the summit.”

• This article was based in part on wire service reports.

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