In his New Year's address, the North Korean dictator said he was willing to send a delegation to next month's Winter Olympics in Seoul, but he also said he had a "nuclear button" on his desk and all US territory is within striking distance of his nuclear weapons.
The US President ridiculed his comments when he boasted about having a bigger and more powerful "nuclear button" than Mr Kim.
“North Korean Leader Kim Jong Un just stated that the ‘Nuclear Button is on his desk at all times.’ Will someone from his depleted and food starved regime please inform him that I too have a Nuclear Button, but it is a much bigger & more powerful one than his, and my Button works!” Mr Trump tweeted.
North Korean Leader Kim Jong Un just stated that the “Nuclear Button is on his desk at all times.” Will someone from his depleted and food starved regime please inform him that I too have a Nuclear Button, but it is a much bigger & more powerful one than his, and my Button works!— Donald J. Trump (@realDonaldTrump) January 3, 2018
The two leaders have exchanged a series of bellicose insults in recent months, with Mr Trump at times dismissing the prospect of a diplomatic solution to a crisis in which North Korea has threatened to destroy the United States.
However, the recent thaw in relations between North and South Korea may show a shared interest in improving ties, although there is no guarantee it will ease tensions.
The North reopened the hotline at the truce village of Panmunjom at 6.30am, when South Korean officials at the border received a call from the North.
During their 20-minute communication, officials of the two Koreas exchanged their names and examined their communication lines to make sure they were working, Seoul's Unification Ministry said.
The hotline with the South was shut down by North Korea in February 2016 in retaliation against the closing of a border factory town in Kaesong that was jointly operated by the two Koreas.
"We will try to keep close communications with the south Korean side from sincere stand(sic) and honest attitude, true to the intention of our supreme leadership, and deal with the practical matters related to the dispatch of our delegation," the North's KCNA news agency quoted Ri Son-gwon, chairman of North Korea's Committee for the Peaceful Reunification of the Fatherland, as saying.
The talks would aim to establish formal dialogue about sending a North Korean delegation to the Olympics in February, Mr Ri said.
US officials have voiced scepticism about the possibility of meaningful talks, particularly if they do not take steps towards banning North Korea's nuclear weapons.
Nikki Haley, the US ambassador to the United Nations, warned North Korea against staging another missile test and said Washington was hearing reports Pyongyang might be preparing to fire another missile.
A spokesman for the Chinese Foreign Ministry, Geng Shuang, said both sides should seize the Olympics as an opportunity to improve ties and make concrete efforts towards alleviating tension.
"All relevant sides should grab hold of this positive trend in the Korean peninsula and move in the same direction," Mr Geng told a daily news briefing in Beijing.
North Korea regularly threatens to destroy South Korea, the US and Japan, and says its weapons are necessary to counter US aggression.
The US has around 28,500 troops stationed in the South, a legacy of the 1950-53 Korean War, which ended in a ceasefire rather than a peace treaty.
Additional reporting by agencies