US Defence Secretary Jim Mattis said there was "possibly some good news" on the summit, while White House Press Secretary Sarah Sanders told reporters: "If the meeting takes place on June 12, we will be ready".
Spokeswoman Heather Nauert said Mr Pompeo and South Korean Foreign Minister Kang Kyung-wha reaffirmed their "shared commitment to the denuclearisation of the Korean peninsula" and pledged to coordinate "in all of their efforts to create conditions for dialogue with North Korea".
South Korean President Moon visited U.S. President Donald Trump earlier this month. No matter how painful sanctions may be, analysts say, Kim would be unwilling to give up his nuclear weapons unless an accord left him feeling completely safe without them.
Relations between the U.S. and North Korean became progressively optimistic over the past few months, after Kim Jong-un invited Mr Trump for a meeting in March.
He has since said he is open to reinstating the talks.
Abe said he planned to hold talks with Trump over the telephone soon after returning to Japan on Sunday.
A senior administration official in the U.S. later gave further details, saying North Korea had shown "a profound lack of good faith".
Joe Hagin, the deputy chief of staff for operations at the White House, was scheduled to travel to Singapore this weekend to make preparations.
Summing up their thoughts on Trump's letter, the mystery prof. said: 'See me after class, Donald.
The statement from Pyongyang appeared created to get the summit back on track after Trump cancelled their planned Singapore meeting, citing "tremendous anger and open hostility" in recent statements from North Korea.
After an unusually placative dispatch from North Korean vice-foreign minister Kim Kye-gwan following Trump's surprise letter, which caught even US allies South Korea and Japan by surprise, the United States appears to be willing to return to the negotiating table.More news: Ex-NAACP official Rachel Dolezal charged with welfare fraud
Asked if it was playing games ahead of the summit, the president responded, "Everyone plays games", adding: "We'll see what happens, it could even be the 12th". "It's the game to win the hearts and minds of every country, to convince them that 'No, we're not at fault; it's the other guy who's at fault'". Washington and Pyongyang will blame the other for failure, and we could very well go back to the same set of conditions that brought us to the brink of war past year, or worse: missile and nuclear tests followed by threats of a military strike.
Japanese Prime Minister Shinzo Abe said however he respected and supported the U.S. president's move.
"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", read Trump's letter to Kim.
The Blue House, South Korea's presidential office, said the two leaders held talks for two hours in the truce village of Panmunjom, where they had met last month and made a declaration vowing to improve ties. They announced they would be holding a candlelit vigil Friday evening in response to Trump canceling the summit.
Trump tweeted Friday morning, a day after he withdrew from the June 12 summit.
He said North Korea remained open to resolving issues with Washington "regardless of ways, at any time".
Trump told reporters at the White House that both sides would like the summit to happen.
It also was unhappy about the North's failure to allow global observers to verify the dismantling of the Punggye-ri test site, the staging ground for all six of its nuclear tests.
DIPLOMATS AT WORK Trump's latest about-face sent officials scrambling in Washington. "We will see what happens", said the US President. "If North Korea is serious, then we look forward to hearing from them at the highest levels".
"He wants to get something that's a long-lasting and an actual real solution", she said.