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Donald Trump has said "very productive talks" are ongoing with North Korea over reinstating a summit with leader Kim Jong Un, adding that if successful it will likely take place in Singapore on June 12.

The US president left open the possibility that the meeting could be put off to another date if negotiations do not make sufficient progress.

He tweeted: "We are having very productive talks with North Korea about reinstating the Summit which, if it does happen, will likely remain in Singapore on the same date, June 12th., and, if necessary, will be extended beyond that date."

Mr Trump shocked the international community on Thursday when he pulled out of the historic summit after talks deteriorated.

On Thursday, White House officials had noted that Mr Trump had left the door open with a letter to Mr Kim that blamed "tremendous anger and open hostility" by Pyongyang but also urged Mr Kim to call him.

By Friday, North Korea issued a statement saying it was still "willing to give the US time and opportunities" to reconsider talks "at any time, at any format".

Mr Trump rapidly tweeted that the statement was "very good news" and told reporters that "we're talking to them now".

Confident in his negotiating skills, Mr Trump made a quick decision to accept the sit-down in March, over the concerns of many top aides, and has remained committed, even amid rising concerns about the challenges he faces in scoring a positive agreement.

Asked on Friday if the North Koreans were playing games with their communications, Mr Trump responded: "Everybody plays games. You know that better than anybody."

While the president did not detail the nature of the new US communication with the North on Friday, Defence Secretary Jim Mattis said at the Pentagon, "The diplomats are still at work on the summit, possibility of a summit, so that is very good news."

He characterised the recent back-and-forth as the "usual give and take".

A previously planned trip by White House aides to Singapore this weekend to work on logistics for the trip remained on schedule, said two White House officials.

Secretary of State Mike Pompeo spoke on Friday with a top official from South Korea, whose leaders had appeared to be taken aback when Mr Trump withdrew from the summit.

Spokeswoman Heather Nauert said 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 Korea's government said in a statement released on Saturday that it was relieved about the revived talks for a summit.

Mr Trump's comments on Friday came after days of mixed messages on the summit.

Mr Trump, in his letter to Mr Kim on Thursday, objected specifically to a statement from a top North Korean Foreign Ministry official. That statement referred to vice president Mike Pence as a "political dummy" for his comments on the North and said it was up to the Americans whether they would "meet us at a meeting room or encounter us at nuclear-to-nuclear showdown".

The president then said from the White House that a "maximum pressure campaign" of economic sanctions and diplomatic isolation would continue against North Korea - with which the US is technically still at war - though he added that it was possible the summit could still take place at some point.

A senior White House official said the North had reneged on its promises ahead of the summit, including a pledge to allow international inspectors to monitor its explosive destruction of its nuclear test site.

Mr Trump's aides had warned that merely agreeing to the summit had provided Mr Kim with long-sought international legitimacy and, if Mr Trump ultimately backed out, he risked fostering the perception that the president was insufficiently committed to diplomatic solutions to the nuclear question.

US defence and intelligence officials have repeatedly assessed the North to be on the threshold of having the capability to strike anywhere in the continental US with a nuclear-tipped missile - a capacity that Mr Trump and other US officials have said they would not tolerate.

Russian president Vladimir Putin said from St Petersburg: "If you don't behave aggressively and if you don't corner North Korea, the result that we need will be achieved faster than many would think, and at less cost."