WASHINGTON -- President Donald Trump and Defense Secretary Jim Mattis spoke on Saturday with their South Korean counterparts after the historic meeting between leaders of the two Koreas. Mr. Trump also said in a tweet Saturday morning that "things are going very well" as he prepares for an expected summit with the North's Kim Jong Un.

Mattis and Defense Minister Song Young-moo said they were committed to "a diplomatic resolution that achieves complete, verifiable and irreversible denuclearization" of the North, according to the Pentagon's chief spokeswoman, Dana W. White. Mattis also reaffirmed "the ironclad U.S. commitment" to defend its ally "using the full spectrum of U.S. capabilities. "

Mr. Trump tweeted Saturday that he had "a long and very good talk" with President Moon Jae-in. He also said he updated Japan's prime minister, Shinzo Abe, about "the ongoing negotiations" for an anticipated summit with Kim, tentatively scheduled for May or early June.

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"Things are going very well, time and location of meeting with North Korea is being set," Mr. Trump tweeted.

Just had a long and very good talk with President Moon of South Korea. Things are going very well, time and location of meeting with North Korea is being set. Also spoke to Prime Minister Abe of Japan to inform him of the ongoing negotiations. — Donald J. Trump (@realDonaldTrump) April 28, 2018

Mongolia and Singapore are the final two sites under consideration for the upcoming summit between Mr. Trump and Kim Jong Un, two administration sources told CBS News' Major Garrett. Mr. Trump told reporters Friday that the location had been narrowed down to two spots.

"I will be meeting with Kim Jong Un in the coming weeks, we look forward to that," Mr. Trump said during a news conference with German Chancellor Angela Merkel. Mr. Trump declined to say whether he has met with Kim directly. But he did say they have a "good working relationship.

Moon and Kim have pledged to seek a formal end to the Korean War, fought from 1950 to 1953, by year's end and to rid the Korean Peninsula of nuclear weapons.

Mr. Trump is claiming credit for the Korean summit. Now he faces a burden in helping turn the Korean leaders' bold but vague vision for peace into reality after more than six decades of hostility. Mr. Trump must contend with suspicions about his own suitability to conduct that kind of war-and-peace negotiation and succeed where his predecessors have failed, and whether Kim really is willing to give up the nuclear weapons his nation took decades acquiring.

"It is still unclear whether North Korea still believes that it can have its cake and eat it too," said Victor Cha, who until January had been in the running to become Trump's choice for ambassador to South Korea.

At a White House news conference with German Chancellor Angela Merkel on Friday, Mr. Trump basked in the afterglow of the meeting between Kim and Moon, and said he has a responsibility to try to achieve peace and denuclearization.

"And if I can't do it, it'll be a very tough time for a lot of countries, and a lot of people. It's certainly something that I hope I can do for the world," he said.

Moon and Kim have not specified what steps would be taken to formally end the war or eliminate nuclear weapons. Now the pressure to deliver results, at least on the allies' side, has shifted to Mr. Trump.

The president pushed back against critics who say he's being manipulated by Kim, who has abruptly shifted to diplomacy after last year's full-scale push to become a nuclear power that could threaten the U.S. mainland.

"I don't think he's ever had this enthusiasm for somebody, for them wanting to make a deal," Mr. Trump said in the Oval Office. "We're not going to be played, OK. We're going to hopefully make a deal. The United States in the past has been played like a fiddle."

New Secretary of State Mike Pompeo, who as CIA director met Kim four weeks ago in North Korea, told reporters in Brussels that he got the impression that Kim was "serious" about negotiating on denuclearization because of the Trump-led economic pressure campaign. But Pompeo -- who traveled to the Middle East on Saturday -- added a word of caution: "I am always careful. There is a lot of history here. Promises have been made, hopes have been raised and then dashed."

North Korea has already called a halt to nuclear and long-range missile tests, which has helped dial down tensions significantly.

North Korea was hit with unprecedented economic restrictions during 2017, when the U.S. and North Korean leaders traded threats while Kim pushed his nation to the verge of being able to fire a nuclear-tipped missile at the U.S. mainland.

The diplomatic climate has changed dramatically this year, as Kim has ended his international seclusion, reaching out to South Korea, the U.S., and China.

Mattis has said the U.S. is "optimistic right now that there's opportunity here that we have never enjoyed since 1950" and any progress will be up to the diplomats. He was referring to the year the Korean War broke out.

The fighting, which also involved China, cost hundreds of thousands of lives and ended with the declaration of an armistice, not a peace treaty. That has left the peninsula in a technical state of war for decades.