Mr. Trump said he received "a very nice note" from Mr. Kim on Twitter.
He also says that bilateral trust should be "further strengthened in the future process of taking practical actions".
"I firmly believe that the strong will, honest efforts, and unique approach of myself and Your Excellency Mr. President aimed at opening up a new future between the DPRK [Democratic People's Republic of Korea] and the U.S. will surely come to fruition", the letter states, concluding, "I extend my conviction that the epochal progress in promoting the DPRK-U.S. relations will bring our next meeting forward".
The message came after Secretary of State Michael Pompeo returned from his third visit to North Korea.
North Korea has said eight of the 17 - including Megumi Yokota, who disappeared while heading home from school when she was 13 - have already died, while four never entered the country, an account Japan has dismissed as unreliable.
Ever since Trump's summit with Kim in Singapore, critics have slammed the president for not doing more to challenge Kim, elevating a dictator on the global stage, and touting a pact the two leaders signed which doesn't provide any solid agreement for a denuclearization plan.
North Korea, meanwhile, lashed out at Washington for presenting "gangster-like demands" for complete, verifiable and irreversible denuclearization.
There was no call or explanation from North Korean officials on why they skipped the meeting.
Elon Musk’s ‘kid-size submarine’ is headed to Thailand for cave rescue
Mr Narongsak said discussions regarding the lowest risk rescue attempts were ongoing, but acknowledged that time was against them. Rescuers have been pumping water out of the cave with the goal of letting the boys walk out, rather than having to scuba dive.
But North Korea's delegation failed show up.
"Both the United States and the North are still in discussion over when to meet", the official added.
The letter is dated July 6.
State Department spokeswoman Heather Nauert said the talks will take place Sunday and the U.S.
Trump has expressed an eagerness to tout the recovery of the war dead, telling Fox News that Kim was "giving us back the remains of probably 7,500 soldiers".
Trump has sought to cast his summit with Kim as a major victory, although reports have suggested North Korea is continuing with its nuclear programme and promises of "denuclearisation" remain vague.
Following the North Koreans' no-show at Thursday's meeting, the USA state department said the two delegations had agreed to meet on 15 July.
But skepticism remains over whether North Korea's findings will satisfy Japan.
The source said that Kim did not see any need to hold dialogue with Japan before his summit with South Korean President Moon Jae-in in April, instead favoring relations with the United States, China and Russian Federation.