The two nations have remained in a state of war and without diplomatic relations since the Korean War ended in 1953 without a peace treaty.
"I probably have a very good relationship with Kim Jong-un", Trump was quoted as telling The Wall Street Journal in an interview on Thursday, refusing to confirm whether the two had spoken. "I have relationships with people".
Trump was unsatisfied and asked again, the officials said.
Trump, it has often been reported, likes to match people to jobs he thinks they look qualified for. "I just don't want to comment".
Washington and Pyongyang are in a standoff over North Korea's missile and nuclear programmes, which could be used to target the U.S. and its allies.
Seoul originally offered Friday to Pyongyang holding working-level talks about the dispatch of all DPRK delegations, including athletes and cheering squads, to the 2018 Winter Olympic and Paralympic Games set to run from February to March in South Korea's eastern county of Pyeongchang.
Trump and Kim have traded bellicose rhetoric and crude insults over the previous year, as North Korea has accelerated weapons tests and appears on the cusp of having a nuclear-tipped missile that could strike the US mainland. Kim called the 71-year-old American president "the mentally deranged USA dotard".
The White House has denied and corrected a quote attributed to US President Donald Trump that suggested he had good contacts with North Korean leader Kim Jong-un. It has, however, retained back-channel communications through the North's diplomatic mission at the United Nations in NY. The two sides agreed on the participation of North Korean athletes in the South Korean Olympics. Multiple outlets have also reported that the Trump administration is debating the possibility of pursuing a "bloody nose" strategy - a first, targeted strike to deter North Korea from proceeding with its weapons testing.
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