The shock announcement was made by senior South Korean officials in Washington, who passed on a letter from the North Korean leader.
They said Mr Kim had also agreed to halt nuclear and missile tests and was "committed to denuclearisation".
It appears to be a major breakthrough after months of threats and violence.
The South Korean delegation had held unprecedented talks with Mr Kim in Pyongyang earlier this week.
National security adviser Chung Eui-yong, speaking outside the White House after a meeting with Mr Trump, said the leaders were hoping to meet by May.
"I told President Trump that at our meeting North Korean leader Kim Jong-un said he's committed to denuclearisation," Mr Chung told a news conference.
"Kim pledged that North Korea will refrain from any further nuclear or missile tests."
He added: "President Trump appreciated the briefing, and said he would meet Kim Jong-un by May to achieve permanent denuclearisation."
Ties between the two Koreas have considerably thawed since their joint participation in the Winter Olympics, but many critics expressed scepticism about the North's sincerity.