While admitting that dealing with North Korea was "going to be a process", Mr. Trump said he believed that process would ultimately be "successful".
In regards to his abrupt cancellation of the summit last week, Trump said "we're over that, we're totally over that".
Trump's is the latest administration to inherit a decades-long crisis on the Korean Peninsula, where North Korea technically remains at war with its US -backed rival since their 1950s conflict.
In the latest sign of hostility cooling down but hopes kept in check, Trump said he had unilaterally put a hold on hundreds of new sanctions against the North, without Kim's government even asking.
"The relationships are building and that's very positive", he said.
"We talked about nearly everything".
Trump said that, after Friday's talks, the parties are "totally over that and now we're going to deal and we're going to really start a process".
After exchanging threats and insults since Trump became president a year ago, the United States and North Korea have been trying to set up the summit between their leaders. However, his comments were the most positive from any USA official since Trump abruptly canceled the meeting last week after belligerent statements from the North.
Later Friday, deputy White House press secretary Hogan Gidley confirmed that Trump had read the letter, but he did not reveal its contents.
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The North Korean envoy, a close confidant of Kim Jong-un, arrived in Washington after two days of talks with US Secretary of State Mike Pompeo in NY.
He added: "I may be in for a big surprise, folks".
After the meeting, Trump posed for photos with Kim Yong Chol outside the Oval Office, and they talked amiably at Kim's black SUV before he was driven away. By the next day, he was signaling the event could be back on after a conciliatory response from North Korea.
Trump made the announcement on Friday, following a two-hour-long meeting with North Korean senior official Kim Yong-chol at the White House.
Kim Jong Un, however, has indicated he may indeed be willing to end his nuclear program and has already demolished his only known nuclear testing site last week as a gesture that he sought peace with the U.S. That same day, however, Trump canceled the talks after North Korea issued a fiery statement criticizing Vice President Mike Pence for threatening Kim Jong Un's life. Despite Kim's apparent eagerness for a summit with Trump, there are many doubts that he would fully relinquish his nuclear arsenal, which he may see as his guarantee of survival.
Despite the upbeat messaging in the United States, Kim Jong Un, in a meeting with Russia's foreign minister on Thursday, complained about the USA trying to spread its influence in the region, a comment that may complicate the summit plans.
"I purposely didn't open the letter".
Yet he also said a news conference that hard work remains including hurdles that may appear to be insurmountable as negotiations progress on the US demand for North Korea's complete, verifiable and irreversible denuclearization.
Kim Yong Chul, who till now was under U.S. sanctions, is likely to stay in the country till tomorrow.
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