Mr Trump made the announcement after Tuesday's historic summit with North Korean leader Kim Jong-un in Singapore.
The move is seen as a big concession to North Korea and appeared to take US allies in the region by surprise.
Meanwhile, North Korean state media said Mr Kim had accepted an offer from President Trump to visit the US.
KCNA news agency said Mr Kim had invited Mr Trump to visit Pyongyang "at a convenient time" and Mr Trump had also invited Mr Kim to the United States.
"The two top leaders gladly accepted each other's invitation," KCNA added.