James Mattis, the United States defence secretary, also said the meeting could take place on June 12 as first planned, providing "our diplomats can pull it off".
Kim Jong-un's comments published by the newspaper did not include any mention of his potential meeting with Trump.
Asked Friday if the North Koreans were playing games with their communications, Mr Trump responded: "Everybody plays games". Another photo showed Moon shaking hands with Kim Jong Un's sister, Kim Yo Jong. He said the meeting could even be put back on track for June 12.
The White House has repeatedly offered mixed messages.
And as of Friday morning, an advance team of about 30 White House staffers and State Department officials was still planning to leave for a logistics meeting Southeast Asia on Sunday, according to two people familiar with the planning.
The meeting at a border truce village of Panmunjom came hours after Seoul expressed relief over revived talks for a summit between Trump and Kim following a whirlwind 24 hours in which US President Donald Trump cancelled the highly anticipated meeting before saying it was potentially back on.
Earlier Trump said North Korea's statement that it was still willing to meet despite his cancellation of the summit was "warm and productive". "Sadly, based on the tremendous anger and open hostility displayed in your most recent statement, I feel it is inappropriate, at this time, to have this long-planned meeting", Trump wrote in a letter to Kim released by the White House.
The talks between Mr Trump and Mr Kim, if they were to go ahead, would be focused on ways of denuclearising the Korean peninsula and reducing tensions. He said: "They very much want to do it, we'd like to do it".
"We weren't getting the right signals previously, so hopefully we will in the future", Nauert said.
North Korea's first vice foreign minister Kim Kye Gwan said on Friday that his country still wanted to pursue peace and said it would give Washington more time to reconsider talks.
Sebastian, Vero Beach to get minimal impact from Tropical Storm Alberto
At this point, the Charlotte area could get two to four inches of rain, with some locations getting as much as five inches. But Julie Hilton said people are cancelling because of the weather and room reservations are down about 20 percent.
On Wednesday, Choe Son-hui, another North Korean vice minister, called Vice President Mike Pence a "political dummy" for suggesting the United States could apply the "Libya model" to North Korea if talks failed.
North Korea issued a statement Friday saying it is still "willing to give the USA time and opportunities" to reconsider talks "at any time, at any format".
Trump, in his letter to Kim, objected specifically to a statement from a top North Korean Foreign Ministry official.
Trump blamed "open hostility" from Kim's regime for his decision to call off the talks, and warned North Korea against committing any "foolish or reckless acts". While there are always "high points and low points" in diplomacy, she said, "we hope that the meeting will go forward at some point".
Even if it does move forward, there's now a question of what can be accomplished after North Korea made clear it will not accept the USA demands for complete, verifiable and irreversible denuclearization. The plan followed months of war threats and insults from both leaders.
If Trump remains tough, he will know quickly whether Washington and Pyongyang can truly make history-or whether they are on a path to armed conflict. But Pyongyang recently issued a series of statements calling out the US for demanding its "unilateral" denuclearization and threatening to pull out of the summit.
But the North's Kim Kye Gwan countered that Pyongyang's angry statements were "just a backlash in response to harsh words from the U.S. side that has been pushing for a unilateral denuclearisation".
"Look, we're always going to be prepared and if the meeting takes place on June 12th, we'll be ready", White House press secretary Sarah Sanders told reporters.
"The diplomats are still at work on the summit, possibility of a summit, so that is very good news", Mattis said.