WASHINGTON -- Americans should be concerned about the mental fitness of North Korean leader Kim Jong-un, not their own President Donald Trump, the White House said Wednesday amid a renewed escalation of rhetoric between the countries.

Trump boasted Tuesday that he has a "nuclear button" that is "much bigger and more powerful" than the one Kim has after the North Korean leader warned that a nuclear button sits on his desk at all times.

Critics here have accused the US president of speaking lightly of a serious matter and even questioned his mental stability.

"I think the president and the people of this country should be concerned about the mental fitness of the leader of North Korea," White House press secretary Sarah Sanders told a regular news briefing. "He's made repeated threats, he's tested missiles, time and time again, for years, and this is a president who's not going to cower down and is not going to be weak, and is going to make sure that he does what he's promised to do; and that's stand up and protect the American people."

She rejected the notion that Trump was taunting Kim in his Tuesday tweet, stating, "I think it's just a fact."

"The president's very well aware of how the process works and what the capacity of the United States is. And I can you tell that it's greater than that of North Korea," Sanders added.

Trump and Kim have exchanged a series of insults and threats in the wake of North Korea's repeated testing of nuclear bombs and ballistic missiles capable of carrying them to the US.

Last year the US president threatened to unleash "fire and fury" on Pyongyang if it continued to threaten his country. (Yonhap)