SEOUL — North Korea said Wednesday it will reopen a hot line with the South to discuss attending the Winter Olympics, forging ahead with peace overtures despite taunts from US President Donald J. Trump who said he has a “much bigger” nuclear button than Kim Jong-Un.

The hot line, which was cut by the North in 2016, is to be restored at 0630 GMT after Seoul proposed high-level talks in response to an olive branch from the North’s leader ahead of next month’s Pyeongchang Games.

Mr. Kim’s overtures to the South marked a rare softening in tone, as tensions over its banned weapons program have surged in recent months following a flurry of missile launches and its most powerful nuclear test yet.

Seoul responded with an offer to hold talks on January 9 — the first since 2015 — to discuss “matters of mutual interest” including Mr. Kim’s suggestion that the reclusive nation could participate in the Olympics.

But Mr. Kim’s New Year address also included a warning to the US that he has a “nuclear button” on his table, prompting a furious response from President Trump via Twitter.

“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!” he said.

Mr. Trump’s remarks came as his ambassador to the United Nations Nikki Haley dismissed Seoul’s offer to hold talks, calling it a “band-aid.”

US State Department spokesman Heather Nauert also warned that Mr. Kim “may be trying to drive a wedge of some sort between the two nations — between our nation and the Republic of Korea.”

But the rapprochement seemed to be moving ahead on Wednesday, with Mr. Kim welcoming Seoul’s support for his overtures, according to Ri Son-gwon, the head of North Korea’s agency handling inter-Korean affairs.

CHANNEL
The two countries, which are divided by a Demilitarized Zone since the end of the 1950-53 Korean War, last held high-level talks in 2015 to try to ease tensions.

The hot line, located in the truce village of Panmunjom, remained operational until February 2016, with operators from both countries checking it twice a day.

The channel was shut down when relations deteriorated over a dispute involving the Kaesong industrial complex, which was jointly operated by both countries.

Seoul welcomed Pyongyang’s decision to reopen the hot line as “very significant,” with chief presidential press secretary Yoon Young-chan saying “it creates an environment where communication will be possible at all times.”

South Korean President Moon Jae-In has long favored engagement with the nuclear-armed North, but the Trump administration insists the regime must give up its weapons drive before any negotiations can take place.

Ambassador Haley told reporters that Washington could not take the talks seriously “if they don’t do something to ban all nuclear weapons in North Korea.”

North Korea has shrugged off a raft of new sanctions and heightened rhetoric from Washington as it drives forward with its weapons program, which it says is for defense against US aggression.

Pyongyang claims it needs nuclear weapons to protect itself from a hostile Washington and has striven to create a warhead capable of targeting the US mainland with an atomic warhead.

Mr. Moon on Tuesday welcomed Mr. Kim’s olive branch as a “positive response” to Seoul’s hopes that the Pyeongchang Olympics would be a “groundbreaking opportunity for peace.”

But any rapprochement between the two countries will take place against a backdrop of suspicion, if not outright hostility, by Washington, with Messrs. Trump and Kim exchanging angry insults since the US leader took office a year ago.

Mr. Trump has mocked Mr. Kim as “fat” and a “little rocket man.” Mr. Kim, for his part has described Mr. Trump as a “mentally deranged US dotard.” — AFP

LONDON — This is the year when the biggest political and economic conundrum to face the UK in modern times will be solved, further complicated or even abandoned in all but name.

Within 10 months, the British government and European Union (EU) aim to have an agreement on their divorce and at least the outline of their future trading relationship. They have different ideas about what Brexit should look like and different views on how the talks themselves should be ordered.

Then there are the divisions within each camp. The UK’s governing Conservatives, whose decades-old rift over EU membership was the catalyst for Brexit, remain split over what it really means. The opposition Labour Party is avoiding the question, and the Scottish nationalists don’t want to leave at all.

The EU may be putting its unity before any other priority, but negotiations could reveal conflicting national interests among the remaining 27 members.

So after a year of flawed predictions, here’s a look at several scenarios that just might come to pass in 2018.

THE BASE CASE
While the UK wants the full trade deal done by the time it leaves on March 29, 2019, the EU wants it ready to go by January 2021. This is what EU experts reckon could happen:

After a couple of months of uneventful talks, an agreement is reached on the transition deal that businesses have been crying out for. It’s not legally binding yet, but it’s enough to prevent a mass exodus of companies. In March, trade discussions start.

The UK fights for the City of London, but soon realizes that all the banks have contingency plans anyway and a fair amount of business will still be done in London after the split. The EU won’t budge on its refusal to let the UK keep the best bits of membership and Prime Minister Theresa May’s Conservatives won’t let her make the concessions that would be needed to remain in the EU’s single market.

In October there’s an outline agreement, which is vague, but detailed enough to be clear that Britain is headed for a trade deal that’s a lot like the one Canada struck with the EU.

It will keep tariffs off most goods, but put up barriers at customs and won’t do much for the service industries that make up most of the UK economy. Carsten Nickel of Teneo puts the chances of this scenario at about 60%.

The Irish border is back as a major obstacle. In the end, Ms. May calls the bluff of her Northern Irish allies in the Democratic Unionist Party (DUP) and they accept that in some areas they will have different rules to the rest of the UK to keep the border with the Republic of Ireland open.

Growing popular support for Labour under Jeremy Corbyn makes the DUP reluctant to walk out on Ms. May and let the leader they loathe come to power.

The dreaded cliff-edge scenario has been avoided, but companies start preparing for the trade barriers that lie ahead. No one knows when the detailed trade negotiations will finally end, so businesses are stuck with the outline for a while to come.

THE UPSET
Talks on transition go well enough. Then, by October, it’s clear negotiations on the future trade partnership are failing. The question of how to keep the Irish border open without a customs agreement rears its head again.

This time, it proves impossible for Ms. May to satisfy her three most difficult audiences: the Irish government, which is backed by the EU; the Democratic Unionist Party, which is propping up her government; and the ardent Brexit-backers within her own party.

Despite two weeks of crisis talks with the DUP, no deal is reached. Ms. May’s minority government collapses and an election is called for Thursday Nov. 15. The Conservatives have no time to choose a new leader so against the odds Ms. May takes the party into the short campaign.

On election day, the first in the fall since 1974, voter turnout is down. Labour manages to mobilize the youth vote while elderly Conservatives stay at home on a damp autumn day. Shortly before dawn, Mr. Corbyn is declared the winner and becomes prime minister with a small majority of 20. Once in government, Labour’s policy moves toward closer ties with the EU.

By December, the time for talking is over as Brexit day looms. At an emergency negotiating session on Christmas Eve, the UK’s chief negotiator, Keir Starmer, accepts the EU’s offer of membership of the European Economic Area.

It means Britain will maintain full access to its biggest market for goods and services, but now has to accept rules it has no say in making. It has also failed to “take back control” of immigration, a major issue in the Brexit debate.

THE WALKOUT
Talks on transition take longer than the UK hoped and the start of proper trade discussions is delayed. It soon becomes clear that the services industry is going to be largely left out of the future trade deal.

Businesses squeal and Brexit-backers at home wonder in public why Ms. May agreed to pay a hefty divorce bill in return for such a bare-bones trade deal. Loose ends that weren’t properly tied up in the first few months of talks continue to dog discussions.

The EU again reminds the UK that it needs to find a way to keep the Irish border open after the split, but it won’t give an inch to help find a solution.

The October deadline comes and goes and toward the end of the year talks break down with the UK team reluctantly walking out. Both sides are now hurtling toward a no-deal Brexit in March 2019 unless they can patch things up quickly.

THE WISH
Talks on trade and transition start quickly in January and by February a deal on transition is agreed.

The UK convinces the EU that it should offer it a “Canada plus, plus, plus” deal. That means a broad trade agreement that keeps tariffs off goods and also allows services companies, including banks, to continue to operate across the continent. The EU was divided on financial services but the pragmatists in Europe win the day.

Britain also convinces the EU that it needs to get a fully detailed trade agreement drafted in time for exit day so that it can be signed immediately after Brexit. Businesses and customs officials now have a palatable two years to prepare for the shift to the new arrangement. — Bloomberg