US Secretary of State Mike Pompeo has said preparations for talks with North Korea are "moving in the right direction" but that there is "a great deal of work to do".
He was speaking after talks with one of Kim Jong-un's most high-profile aides, Gen Kim Yong-chol, in New York.
Gen Kim will now travel to Washington to deliver a letter from Kim Jong-un to President Donald Trump, he said.
Mr Trump has also downplayed hopes of reaching a quick deal with Mr Kim.
He said while he would "like to see it done in one meeting... often times that's not the way deals work".
"But it'll get done at some point."
Separately, the North Korean leader has agreed to hold a summit with Russia, after a meeting with Foreign Minister Sergei Lavrov.
North Korean and US officials have been working to get the 12 June Singapore summit back on track after the US leader, angered by North Korean rhetoric, announced last week he was no longer attending.
If it does go ahead, it will be the first time a sitting US president has met a North Korean leader. Pyongyang has long wanted such a meeting, which it would see as validating its status.
The talks will focus on North Korea's announcement earlier this year that it is committed to "denuclearisation" - but there is as yet no agreement between the two sides about what that might mean and how it would be achieved.
Gen Kim - who is vice-chairman of North Korea's ruling Workers' Party's Central Committee - is the most senior North Korean to visit the US in nearly 20 years.
At his press conference on Thursday, Mr Pompeo said they had "made real progress" towards bringing their leaders to the summit.
The US-North Korea relationship was at a "pivotal moment", he said, and it would be "nothing short of tragic to let this opportunity go to waste".
"This is going to be a process that will take days and weeks to work our way through," he said.
What is at stake?
The US has long called on North Korea to abandon its nuclear weapons programme but Pyongyang maintains it needs the weapons for its security.
In exchange for North Korea giving up its atomic arsenal, the US could possibly relax sanctions, offering food and other aid to the impoverished country.
However, Pyongyang has strongly objected to statements by Trump administration officials comparing North Korean denuclearisation to Libya's.
Former Libyan leader Colonel Gaddafi gave up his nuclear programme only for him to be killed by Western-backed rebels a few years later.
Who else has been talking?
In North Korea, Mr Kim held a meeting with Mr Lavrov and, according to state media, told him that the "will for denuclearisation of the Korean peninsula still remains unchanged and consistent and fixed".
The BBC's Laura Bicker in Seoul says Mr Kim's agreement to meet President Vladimir Putin in Russia later this year has sent a signal to Washington that North Korea has powerful allies.
North Korea's Deputy Foreign Minister Choe Son-hui also met Sung Kim, a former US ambassador to South Korea, at Panmunjom on the border between the two Koreas.
In Singapore, where the summit will be held, a US team led by White House official Joe Hagin is thought to be meeting Kim Chang-son, the de facto chief of staff to the North Korean leader, to talk about logistics.
US Defence Secretary Jim Mattis is also travelling to Singapore for an annual Asian security summit.
He told reporters on his plane on Thursday he would reassure US allies the White House was committed to their regional security.