(CNN) Talks between US President Donald Trump and North Korean leader Kim Jong Un in Singapore could go into a second day if they're going well, sources told CNN.

US officials have laid the groundwork for a second day of meetings if the two men decide they want to continue discussions, according to two sources familiar with the matter.

Trump and Kim are due to meet on June 12 for the first time and the US leader is due to return home the next day, but US officials in Singapore have established a contingency plan for a second day of discussions.

A White House spokesman declined to comment on the matter.

'Getting to know you'

It's not clear if Trump is eager for a two-day summit with Kim, but he has expressed a desire for flexibility in the negotiations and has stressed to aides and US allies that he plans to follow his gut as he negotiates with the North Korean leader.

The summit will be a historic first meeting between a sitting US president and North Korean leader. Talks will begin at the Capella Hotel on Singapore's Sentosa Island at 9 a.m. local time on June 12 (9 p.m. June 11 ET).

The US has stressed that it is seeking complete and verifiable denuclearization of North Korea, but a week ahead of the summit, questions remain about whether Kim is truly committed to achieving that aim. US officials emphasize that it is a decision Kim still needs to make.

While leaving open the possibility of a second day of meetings, US officials have been wary of over hyping expectations for what can be accomplished during Trump and Kim's first meeting.

The President joined in that effort last week when he made clear the meeting would be the beginning of a dialogue with North Korea and that it was unlikely he would achieve all his objectives in a first meeting.

"I think it's a 'getting to know you' meeting plus, and that could be a very positive thing," Trump said.

Trump and his aides have suggested the most concrete product of the June 12 encounter could be a peace agreement formally ending the Korean War -- a far cry from the commitment to immediate denuclearization that the administration once insisted would be required for Trump to come to the table.

Now, the most some aides are hoping for is a broad declaration from Kim that he's open to getting rid of his weapons, with the work of securing specific commitments from the North Korean regime handed off to aides for a process that experts predict could take years.

On and off summit

The meeting itself is an stunning turn of events after months of antagonism and brinksmanship between the US and North Korea over Pyongyang's nuclear missile program.

It almost didn't happen after Trump called off the talks at the last minute in May, citing "open hostility" from Pyongyang.

That came after an angry rebuke from North Korea in response to suggestions by Trump's national security adviser John Bolton that the country could follow the "Libya model" of denuclearization.

Within days however, the talks were back on, and Bolton has since been sidelined by Secretary of State Mike Pompeo, who has been leading negotiations with North Korea, according to sources familiar with the matter.

Another Trump aide waded into the matter Wednesday, with Rudy Giuliani, the President's personal lawyer, saying Kim Jong Un had begged on his "hands and knees" for the summit to go ahead.

North Korea has not yet reacted to Giuliani's comments.