Trump has derided the North Korean leader as a "maniac" and referred to him as "little rocket man." Kim has responded by calling the US president a "mentally deranged US dotard."
"I probably have a very good relationship with Kim Jong Un," Trump said in an interview with The Wall Street Journal. "I have relationships with people. I think you people are surprised."
Kim has warned the United States that he intends to build a nuclear arsenal capable of hitting the United States, prompting threats of military action by Washington.
Asked whether he has spoken with the North Korean leader, Trump told the newspaper: "I don't want to comment on it. I'm not saying I have or haven't. I just don't want to comment."
In November, Trump said while on a trip to Vietnam that becoming friends with Kim "might be a strange thing to happen but it's a possibility."
Kim, in a speech last week, said the "nuclear button is always on my desk," prompting Trump to respond in a tweet that his nuclear button is "a much bigger & more powerful one than his, and my Button works!"
"You'll see that a lot with me," he said, "and then all of the sudden somebody's my best friend. I could give you 20 examples. You could give me 30. I'm a very flexible person."
Trump told the newspaper a decision by the United States and South Korea to postpone military exercises until after next month's Winter Olympics in South Korea "sends a good message to North Korea."
North and South Korea held their first talks in two years on Tuesday. Trump told a news conference in Washington on Wednesday the United States would be willing to speak to Pyongyang "under the right circumstances."
(Reporting by Makini Brice; Writing by Katanga Johnson; Editing by Tim Ahmann and Leslie Adler)
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)