U.S. President Donald Trump says he is still looking at a June 12 summit date with North Korean leader Kim Jong Un in Singapore, just days after the U.S. leader said he was canceling the historic meeting.

"We're doing very well in terms of the summit with North Korea," Trump said at the White House on May 26.

"It's moving along very nicely. So, we're looking at June 12th in Singapore. That hasn't changed. So, we'll see what happens," he said.

He added that there is a "lot of good will," and denuclearization of the Korean Peninsula would be "a great thing."

Meanwhile, South Korean President Moon Jae-in, who met with Kim in a surprise summit earlier on May 26, told reporters that Washington and Pyongyang are to planning hold "practical talks" soon on the June 12 summit.

Moon said Kim reaffirmed his commitment to "complete" denuclearization of the Korean Peninsula and to a meeting with Trump.

"He [Kim] also expressed his intention to put an end to the history of war and confrontation through the success of the North-U.S. summit and to cooperate for peace and prosperity," Moon added.

Trump had called off the summit in a letter to Kim on May 24, citing North Korea's threat the day before to cancel the summit in a statement condemning Vice President Mike Pence as a "political dummy."

"Sadly, based on the tremendous anger and open hostility displayed in your most recent statement, I feel it would be inappropriate, at this time, to have this long-planned meeting," Trump wrote in his letter to Kim.

But he left the door open to a new meeting, saying in the letter to Kim, "If you change your mind having to do with this most important summit, please do not hesitate to call me or write."

After their surprise summit at the Panmunjom border village on May 26, the two Korean leaders said they had agreed to meet "frequently" and suggested another face-to-face encounter was likely on June 1.

"They shared the opinion that they would meet frequently in the future to make dialogue brisk and pool wisdom and efforts, expressing their stand to make joint efforts for the denuclearization of the Korean peninsula," North Korea’s KCNA news agency reported.

It was their second summit in the past two months. After the meeting, KCNA said that another “high-level” meeting would be held on June 1, but it did not specifically say that the two Korean leaders would participate.

KCNA suggested following the summit that Kim was still interested in meeting with the U.S. president.

"Kim Jong Un thanked Moon Jae-in for much effort made by him for the DPRK-U.S. summit scheduled for June 12, and expressed his fixed will on the historic DPRK-U.S. summit talks," the report said, using the abbreviation for North Korea.

South Korea's presidential spokesman responded that "we are cautiously optimistic that hope is still alive for U.S.-North Korea dialogue."

With reporting by AP, dpa, AFP, and Reuters