He continued to note that the US has no current plans to relax sanctions on North Korea until there is evidence the country is taking steps to denuclearize, saying the Trump administration would continue its campaign of "maximum pressure".
But the North Korea statement separates Mr. Trump from "high-level officials", thus allowing the the country to rail against the US, while leaving the door open for the president to continue his dialogue with leader Kim Jong Un.
Mr Trump and North Korean leader Kim Jong-un agreed at the June meeting to work towards denuclearisation. Among the agreements was holding another inter-Korean summit in the fall in Pyongyang.
The United States is "not willing to wait for too long" for North Korea to take steps toward denuclearization, U.S. Ambassador to the United Nations Nikki Haley said on Wednesday.
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North Korea remains unchanged in its will to implement agreements made by the top leaders of North Korea and the United States during their Singapore talks, the Korean Central News Agency (KCNA) quoted a foreign ministry spokesperson as saying on Thursday.
"We're going to continue to apply maximum pressure to North Korea until they denuclearize, just as we are to Iran".
The United States was "attempting to invent a pretext for increased sanctions against the DPRK".
With such moves, "one can not expect any progress in the implementation of North Korea-US joint statement including the denuclearization, and furthermore, there is no guarantee that the hard-won atmosphere of stability on the Korean Peninsula will continue", said the spokesperson.
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The White House and US State Department did not immediately respond to requests for comment.
"The U.S. should, even at this belated time, respond to our honest efforts in a corresponding manner", North Korea added in its statement.
"This is all in North Korea's court", Haley told reporters travelling with her during a visit to Colombia. North Korea also accused unidentified high-level USA officials of "going against the intention of President Trump" by "making baseless allegations against us and making desperate attempts at intensifying the worldwide sanctions and pressure".
The North apparently wants it to ensure the security of its regime.
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The report said: "North Korea has not stopped its nuclear and missile programs and continued to defy Security Council resolutions".