"We'll see what happens. It could even be the 12th. We're talking to them now. They very much want to do it, we'd like to do it," he told reporters.
On Thursday, he blamed the North's "open hostility" for the cancellation of his talks with Kim Jong-un.
North Korea later said it was willing to talk "at any time in any form". Vice-Foreign Minister Kim Kye-gwan said Mr Trump's cancellation decision was "extremely regrettable".
|President Donald Trump said: "We'll see what happens. It could even be the 12th. We're talking to them now. They very much want to do it, we'd like to do it".
The 12 June summit in Singapore would have been the first time a sitting US president had met a North Korean leader.
The details of the now scrapped talks were unclear. But they would have focused on ways of denuclearising the Korean peninsula and reducing tensions.
Just hours before Mr Trump cancelled the summit, North Korea said it had carried out its promise to dismantle tunnels at its only nuclear test site.
What were Mr Trump's latest comments?
Speaking to reporters outside the White House in Washington, the US president said: "We gonna see what happens. We're talking to them [North Korea] now. It was a very nice statement they put out."
And Mr Trump added: "Everybody plays games."
Shortly before this, Mr Trump welcomed North Korea's willingness to hold talks "at any time", describing it as "warm and productive".
Separately, US Defence Secretary James Mattis echoed Mr Trump's comments.
"We have got... possibly some good news on the Korea summit, where it may, if our diplomats can pull it off, may have it back on," he told reporters.
What about Mr Trump's cancellation letter?
It was personally addressed to Mr Kim.
Mr Trump said he had been "very much looking forward" to meeting Mr Kim.
"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 the long-planned meeting," Mr Trump said in a letter to Mr Kim.
"You talk about your nuclear capabilities, but ours are so massive and powerful that I pray to God they will never have to be used," he added.
But he called the meeting a "missed opportunity", saying "some day, I look very much forward to meeting you".
In a separate statement at the White House, Mr Trump said the step was a "tremendous setback for North Korea and the world", adding the US military was "ready if necessary" to respond to any "reckless" act from North Korea.
A senior administration official in the US later gave further details, saying North Korea had shown "a profound lack of good faith".
There were a series of "broken promises" from Pyongyang, the official told reporters, including when the White House sent the deputy chief of staff to Singapore to meet North Korean diplomats ahead of the summit.
What Mr Trump was referring to?
He was apparently responding to statements from a senior North Korean diplomat attacking his administration and casting doubt over the meeting.
Choe Son-hui had said the suggestions from US Vice-President Mike Pence that North Korea "may end like Libya" was "stupid".
Ms Choe, who has been involved in several diplomatic interactions with the US over the past decade, said the North would not "beg" for dialogue and warned of a "nuclear showdown" if diplomacy failed.
A White House official quoted by Reuters news agency described the comments about Mr Pence as the "last straw". They stressed, however, there was a "backdoor that's open still".
References to Libya have angered North Korea. There, former leader Colonel Gaddafi gave up his nascent nuclear programme only for him to be killed by Western-backed rebels a few years later.
North Korea says that, unlike Libya, it is a fully fledged nuclear state. It is insistent it will not engage in any peace process that jeopardises its leadership or its survival as a state.
What's the reaction been?
South Korean President Moon Jae-in said he was "very perplexed" and that it was "very regrettable" that the summit was not going ahead.
He was not informed of the decision before Mr Trump's announcement, reports said.
It was South Korean officials who first informed the US earlier this year that Mr Kim was prepared to discuss potential nuclear disarmament.
In April, the leaders of both Koreas had a historic meeting at the border,promising to end hostilities and work towards the denuclearisation of the Korean peninsula.
UN Secretary General Antonio Guterres said the US and North Korea should not give up, saying "nerves of steel" were required.
In the US, Republican Senator Tom Cotton praised President Trump for "seeing through Kim Jong-un's fraud". But Democratic Senator Brian Schatz said the move was what happened "when amateurs are combined with warmongers".