He reopened a communications channel closed two years ago, and suggested he might send North Korean athletes to the upcoming Winter Olympics in South Korea.
Trump had tweeted last week: "Does anybody really believe that talks and dialogue would be going on between North and South Korea right now if I wasn't firm, strong and willing to commit our total "might" against the North".
He said the US would get involved "at the appropriate time".
The repeated weapons tests earned the North toughened United Nations sanctions, and Kim and Trump exchanged threats of nuclear war and crude personal insults. They'd be doing no talking or it would be much more serious.
Unification Minister Cho Myoung-gyon will meet on Tuesday his quasi-counterpart from North Korea, Ri Son-gwon, who chairs the Committee for the Peaceful Reunification of the Fatherland.
The 'Truce Village' between North and South Korea
It's yet another sign that North Korea's leader Kim Jong Un is looking for ways to calm tensions ahead of the 2018 Winter Olympics in South Korea next month.
Kim, in his New Year's Day address, had said North Korea's nuclear weapons can reach anywhere in the U.S. and said he had a nuclear button on his desk.
Tillerson said how such talks might begin had yet to be determined, but North Korea would have to signal it understood that they would be aimed at "final, full denuclearisation".
But as things go on a different tangent from what the United States expected in the Korean peninsula, President Trump seems to be now trying to "steal" the credit for the talks.
Trump's tweet was in response to Kim's comments in his annual New Year's Day address that "the entire mainland of the U.S.is within the range of our nuclear weapons and the nuclear button is always on the desk of my office. What we're trying to do is make sure we don't repeat what's happened the last 25 years", she said.
Relations between North Korea and the worldwide community have been tense in recent months following a slew of intercontinental ballistic missile launches.