On Wednesday, North Korea failed to show up as scheduled in Panmunjom to discuss the process of repatriating the remains of U.S. soldiers killed in the Korean War.
Kim expresses "invariable trust and confidence" in the president and wishes that "epochal progress" in promoting relations will "bring our next meeting forward".
State Department spokeswoman Heather Nauert said North Korea called at midday Thursday and offered to meet on Sunday instead.
Kim also said he "deeply" appreciates Trump's "energetic and extraordinary efforts" to improve relations and sign a joint statement calling for denuclearization of the Korean Peninsula.
As far as it goes as a negotiating tactic, North Korea's reluctance to show up for scheduled talks is well-documented.
North Korea last repatriated the remains of six individuals in 2007.
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Just hours after U.S. Secretary of State Mike Pompeo left Pyongyang, the North's foreign ministry openly criticized the U.S. for making what it said was a "unilateral and gangster-like" demand for "complete, verifiable and irreversible" denuclearization (CVID).
Separately, the USA has accused North Korea of violating a United Nations sanctions cap on refined oil products.
Nevertheless, Trump on Thursday praised Kim, saying that "great progress" was being made in negotiations. That's according to documents seen by The Associated Press on Thursday and sent by the U.S.to the Security Council committee monitoring the sanctions.
"I think the USA will make a decision in July, hold consultations in August and declare an end to the Korean War in September", said Mr Hong Min, a senior researcher at the Korea Institute for National Unification.
The U.S. had already sent a hundred wooden caskets to Panmunjom to carry the soldiers' remains back home, . once an agreement was reached. He later said at a Cabinet meeting that the remains were in the process of being returned.
More importantly, the letter from Kim, written for Trump before it became clear to North Korea that Pompeo's visit would be an unproductive farrago, is full of flattery.
"North Korea would have to deal with the consequences of breaking away from the negotiations if it were to do so, so it will likely not make an extreme decision", Kim said.