"He speaks and his people sit up at attention. I want my people to do the same," the president told the Fox News Channel program "Fox & Friends."
It is unclear whether Trump referred to Americans broadly or executive branch staff.
The president later said he was "kidding" and questioned whether reporters understood sarcasm.
His comments came as he talked about his relationship with Kim after a summit between the pair on nuclear disarmament. At the meeting, they signed a loose commitment to denuclearization.
Trump has faced backlash for his comments about Kim, whose regime has overseen prison camps, widespread malnutrition and violence against political opponents including family members. Since the summit, Trump has called Kim "tough" for the way he runs his country. He has labeled the dictator "talented" and "smart," saying Kim "loves his people."
Trump also has said North Koreans "love" Kim. Critics of his comments have noted that people who oppose the North Korean regime face punishment.
The president defended his flattering comments about Kim on Friday as part of his effort to get North Korea to abandon its nuclear ambitions.
"You know why? Because I don't want to see a nuclear weapon destroy you and your family," the president told reporters when asked why he compliments Kim.
Trump also said Friday that he will "be calling North Korea" on Sunday.