But the letter appeared to come a day before North Korea accused USA diplomats of "gangster-like" negotiating tactics in hammering out the details of the broad agreement Trump and Kim signed at the summit in Singapore on June 12.
Trump described the letter as a "very nice note" and said, "Great progress being made!".
The release of the letter comes amid growing skepticism around Pyongyang's willingness to denuclearize.
South Korean newspaper Chosun Ilbo reported last week that Secretary of State Mike Pompeo brought two gifts when he met with North Korean officials in Pyongyang: a letter to Kim from Trump, and an Elton John CD with the song "Rocket Man" signed by Trump.
The Trump administration will be ready, Nauert said.
"President Trump still believes that the agreement that was reached in Singapore where North Korea agreed to complete denuclearization represents a commitment that we can make progress on and we are going to continue to work through that", the vice president said.
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But in stark contrast, Pyongyang's take was overwhelmingly negative, with the North warning that the future of the peace process was being jeopardized by "unilateral and gangster-like" United States demands for its nuclear disarmament.
In another apparent setback, the USA accused Pyongyang of violating sanctions on refined oil products and ordered the United Nations security council to immediately halt all related deliveries to North Korea.
Kim asked China to help ease worldwide sanctions against North Korea during his most recent visit to Beijing in mid-June, according to Japanese newspaper Yomiuri Shimbun.
Trump has touted the return of remains as one of the successes of his historic June 12 summit with Kim in Singapore. Faced with mid-term elections in November, Trump has no choice but to put the most positive construction on these debacles, and Kim now has China's backing and appears to fear nothing. The repatriation under discussion now would be the first since the work of U.S. military experts and North Korean workers from 1996 to 2005 who recovered remains believed to be those of more than 220 U.S. soldiers.
Pompeo has insisted that a raft of tough economic sanctions imposed on North Korea would remain in place until "final, fully verified denuclearization" occurs.
"If North Korea gives more substance on the implementation of denuclearisation, and if South Korea and the United States quickly take comprehensive corresponding measures, the whole process will accelerate". The talks Sunday are expected to involve officials from the Pentagon and the U.N. Command, which commanded USA -led allied forces during the war and is involved in maintaining the armistice that ended the fighting in 1953.
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.