White House spokeswoman Sarah Sanders told reporters Trump did not want a meeting that was "just a political stunt".
"He wants to get something that’s a long-lasting and an actual real solution. And if they are they are ready to do that then ... we’re certainly ready to have those conversations," she said.
"If the meeting takes place on June 12, we'll be ready, if it takes place on July 12, we will be ready."
Sanders indicated it had yet to be decided whether White House officials would stick to a plan to travel to Singapore in coming days to prepare the summit.
North Korea had sharply criticised suggestions by Trump's national security adviser, John Bolton, and Vice President Mike Pence that it could share the fate of Libya if it did not swiftly surrender its nuclear arsenal. Libyan leader Muammar Gaddafi was deposed and killed by NATO-backed militants after halting his nascent nuclear program.
"TRUMP FORMULA"
Trump had initially sought to placate North Korea, saying he was not pursuing the "Libya model" and Sanders said he was following "the President Trump model."
US regional allies Japan and South Korea, as well as North Korea's main ally, China, urged the two countries to salvage the summit on Friday.
At an economic forum in St. Petersburg, Chinese Vice President Wang Qishan said it was necessary to ensure security on the Korean peninsula, which touched on China's core interests.
Japanese Prime Minister Shinzo Abe, at the same forum, regretted the cancellation and said the world should keep doing its bit to make the summit happen.
South Korea also would continue efforts to improve ties with the North, the office of President Moon Jae-in said after Moon's top security advisers met for the second time on Friday.
Some analysts worried that cancelling the summit could prompt a resumption in hostilities, including renewed shorter-range missile tests or stepped-up cyber attacks by Pyongyang and increased sanctions or deployment of new military assets by Washington.
In his letter, Trump warned Kim of the United States' greater nuclear might, reminiscent of his tweet last year asserting that he had a "much bigger" nuclear button than Kim.
While the Trump administration had insisted on North Korea's complete, verifiable and irreversible dismantling of its nuclear program, Pyongyang had always couched its language in terms of denuclearisation of the Korean peninsula.
It has said in previous, failed talks that it could consider giving up its arsenal if the United States provided security guarantees by removing its troops from South Korea and withdrawing its so-called nuclear umbrella of deterrence from South Korea and Japan.
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