Tillerson said that eventually there would be a day when the USA and the North decide it's a good time to have that first conversation that would lead to negotiations.
At a news conference in Vietnam on Sunday after his tweets, Trump said it was possible he could be friends with Kim one day and that it would be "very, very nice" but he was not sure that it would happen.
The North Korean foreign ministry reportedly called Trump's Asia trip a "a warmonger's trip for confrontation with our country".
"Reckless remarks by an old lunatic like Trump will never scare us or stop our advance", said the ministry.
On Saturday Night Live, "Weekend Update" host Michael Che couldn't help but tackle the most-talked-about tweet of the night.
Shortly after that speech, North Korean officials told CNN, "We don't care about what that mad dog may utter", referring to Trump.
President Donald Trump and North Korean leader Kim Jong Un engaged in a new round of and jibes Saturday, when Trump sarcastically said he would never call Kim "short and fat" after North Korea described Trump as a "dotard".
For the last months, the two leaders have been on two ends of a bitter war of words prompted in part by Pyongyang refusal to cease ballistic missile tests and the development of nuclear capabilities.
"Unusual things happen in life", Trump said.
Those comments drew sharp criticism in Washington, where Rep. Adam Schiff (D-Calif.), the ranking member of the House Permanent Select Committee on Intelligence, said the president understands the truth about Russian interference, and is simply choosing to accept Putin's denials "over the solid evidence of our own intelligence agencies".
Town told ABC News that if Pyongyang conducted a test during or following Trump's trip, it would likely be with a Hwasong-14, what the USA refers to as the KN-20 - a two-stage intercontinental ballistic missile (ICBM).
This is the first time three carrier strike groups will operate together since 2007 during an exercise off the coast of Guam, according to the Navy.
We'll see what happens now.
But the administration shouldn't be too quick to praise North Korea's restraint.
The secretive kingdom's state-controlled media service told the North Korean people today how their former countrymen south of the Demilitarised Zone are suffering "untold pains under the USA occupation and domination".