North Korean leader Kim Jong Un has reopened a key cross-border communication channel with South Korea for the first time in nearly two years as the rivals explore the possibility of talks after months of acrimony.

Mr Kim’s announcement, read by a senior official on state TV, followed a South Korean offer on Tuesday of high-level talks to find ways to co-operate on next month’s Winter Olympics in the South and discuss other inter-Korean issues.

Ri Son Gwon, chairman of the state-run Committee for Peaceful Reunification, said Mr Kim welcomed South Korea’s overture and had ordered officials to reopen a communication channel at the border village of Panmunjom.

South Korea quickly welcomed the decision and later confirmed the two Koreas had started preliminary contacts on the channel.

But even among the sudden signs of easing animosity, Donald Trump threatened Mr Kim with nuclear war.

In his new year address on Monday, Mr Kim said he was willing to send a delegation to the Winter Olympics, but he also said he has a “nuclear button” on his desk and that all US territory is within striking distance of his nuclear weapons.

This prompted the US president to boast of a bigger and more powerful “nuclear button” than Mr Kim’s – despite the fact that the US president does not actually have a physical button.

The two leaders exchanged crude insults last year after the UN imposed new sanctions over the North’s sixth and most powerful nuclear test explosion and a series of intercontinental ballistic launches.

In the latest escalation of their rhetoric, Mr Trump wrote on Twitter: “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!”

Despite ratcheting up the tension, Mr Trump does not have a nuclear button.

The process for launching a nuclear strike is secret and complex and involves the use of a nuclear “football”, which is carried by a rotating group of military officers everywhere the president goes and is equipped with communication tools and a book with prepared war plans.

If the president were to order a strike, he would identify himself to military officials at the Pentagon with codes unique to him.

The codes are recorded on a card known as the “biscuit” which is carried by the president at all times.

He would then transmit the launch order to the Pentagon and Strategic Command.

Speaking about North Korea, Mr Trump also said the US-led campaign of sanctions and other pressure were beginning to have a “big impact” on the country.

He referred to the recent, dramatic escape of at least two North Korean soldiers across the heavily militarised border into South Korea.

“Soldiers are dangerously fleeing to South Korea. Rocket man now wants to talk to South Korea for first time. Perhaps that is good news, perhaps not - we will see!” Mr Trump said, using his derisive nickname for the young North Korean leader.

The recent softening of contact between the rival Koreas may show a shared interest in improved ties, but there is no guarantee tensions will ease.

There have been repeated attempts in recent years by the rivals to talk, but the efforts often end in recrimination and stalemate.

Outside critics say Mr Kim may be trying to use better ties with South Korea as a way to weaken the alliance between Washington and Seoul as Pyongyang grapples with toughened international sanctions over its nuclear and missile programmes.

Since taking office last May, South Korea’s liberal President Moon Jae-in has pushed hard to improve ties and resume stalled co-operation projects with North Korea. Pyongyang had not responded to his outreach until Mr Kim’s new year address.

Relations between the Koreas soured under Mr Moon’s conservative predecessors, who responded to the North’s expanding nuclear programme with hardline measures.

All major rapprochement projects were put on hold one by one, and the Panmunjom communication channel has been suspended since February 2016.

Mr Moon has pushed for more pressure and sanctions on North Korea, but he still favours dialogue as a way to resolve the nuclear stand-off.

The Trump administration says all options are on the table, including military measures against the North, but Mr Moon has repeatedly said he opposes any war on the Korean peninsula.

Some observers believe these differences may have led Mr Kim to think he could drive a wedge between Seoul and Washington as a way to weaken their alliance and international sanctions.

Talks could provide a temporary thaw in strained inter-Korean ties, but conservative critics worry that they may only earn the North time to perfect its nuclear weapons.

After the Olympics, inter-Korean ties could become frosty again because the North has made it clear it has no intention of accepting international calls for nuclear disarmament and instead wants to bolster its weapons arsenal in the face of what it considers increasing US threats.