Mr Trump tweeted that he had a “much bigger & more powerful” nuclear button after Kim warned “the whole territory of the US is within the range of our nuclear strike”.
Mrs May, who was visiting homeowners who had benefited from stamp duty reforms in Wokingham, was asked whether she was “concerned about this rhetoric” from President Trump.
She said: “We’ve consistently said that the programme that North Korea has in relation to nuclear weapons and ballistic missiles is reckless.
“This is illegal and we are working with the international community, including with the United States, to bring as much political and economic pressure to bear on North Korea to see them change course.”
Earlier on Tuesday, President Trump sounded open to the possibility of an inter-Korean dialogue after Mr Kim made a rare overture towards South Korea in his new year address.
But President Trump’s ambassador to the United Nations insisted talks would not be meaningful unless the North was getting rid of its nuclear weapons.