"I know that a letter is being delivered to me, a personal letter from Kim Jong Un to me, that was handed at the border," Trump told reporters traveling with him to North Dakota.
"I think it's going to be a positive letter."
Trump, who said he expects Secretary of State Mike Pompeo to deliver the note, also brought up a statement from Kim on Thursday emphasizing his continuing "trust in Trump," despite difficulties in denuclearization negotiations.
"That was a very positive statement, what he said about me," Trump said. "There's never been a more positive statement."
He accused the news media of not covering it, however, adding: "Honestly I didn't see it on the front page of your papers."
Trump lauded the progress he said had been made with Pyongyang since the June summit with the North Korean leader in Singapore.
"We have our hostages back," he said. "I say it a hundred times -- no missiles, no rockets, no nuclear testing. There's been some rhetoric, let's see what happens."
Kim on Thursday renewed his commitment to the goal of denuclearization in talks with a special envoy from Seoul, ahead of a summit planned in Pyongyang September 18-20 with South Korean President Moon Jae-in.
"The north and the south should further their efforts to realize the denuclearization of the Korean peninsula," KCNA quoted Kim as saying.
Kim's expression of confidence in Trump, relayed by Seoul's envoy, prompted the US president to tweet out his thanks to the North Korean leader and vow to "get it done together."
Trump and Kim pledged to denuclearize the Korean peninsula at the Singapore summit but no details were agreed.
And Washington and Pyongyang have sparred since on what that means and how it will be achieved.
'Still more work to do'
Trump in late August canceled a planned trip to Pyongyang by Pompeo over a lack of progress in North Korea's nuclear disarmament.
But relations seem to have improved since then.
South Korean national security advisor Chung Eui-yong, who met with Kim, said the North Korean leader also emphasized that his "trust in Trump remains unchanged," the comment which led to the US president's tweet.
Chung added that Kim expressed his intention to work closely with the United States to achieve denuclearization "in the first official term of President Trump," which ends in January 2021.
In a statement on August 24, Trump said he was scotching Pompeo's trip "because I feel we are not making sufficient progress with respect to the denuclearization of the Korean Peninsula."
He also slammed China as not helping with the effort to convince Pyongyang to halt its nuclear weapons and ballistic missile program.
In New Delhi on Thursday, Pompeo, the former US intelligence chief who heads the US negotiating effort, struck a sober note, saying there was still much work to do.
North Korea "is the only country that has commitments under UN Security Council resolutions," Pompeo told reporters.
"It is the case that there is still an enormous amount of work to do. We haven't had any nuclear tests, we haven't had any missile tests, which we consider a great thing," Pompeo added.
"But the work of convincing Chairman Kim to make the strategic shift which we talked about, for a brighter future for the people of North Korea, continues."
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