But the prospects for the office's opening were quickly plunged into doubt as their high-stakes meeting ended without any agreement. Trump told reporters he has not committed to a third summit with Kim.
If the office opened, it would be the first of its kind between the wartime foes and could be a preliminary step toward normalizing their relations once the U.S. feels North Korea no longer poses any significant threat.
Here's a look at what opening a liaison office could mean for ties between the two countries:
LEADERS' WELCOME OFFICE
Asked by an American reporter at their second summit in Hanoi if he was willing to allow the U.S. to open an office in Pyongyang, Kim said through a translator, "I think that is something which is welcomable."
Trump said that he considers it "a good idea" and that it should happen "both ways," suggesting he wants North Korea to open its own liaison office in Washington. No further details were available.
Many outside experts had said the opening of a liaison office was among possible things Trump could offer to persuade Kim to promise firmer denuclearization measures in Hanoi.
The North has complained about the lack of action taken by the U.S. since their first summit in June, when Kim and Trump agreed to establish new relations and build a lasting peace on the Korean Peninsula. The U.S. has been pushing for North Korea to take substantial disarmament steps before getting any serious concessions.
NO EMBASSIES FOR ENEMIES
The two countries have no embassies in each other's capitals because they have no diplomatic ties. They are actually in a technical state of war because the 1950-53 Korean War ended with an armistice, not a peace treaty.
Sweden, which has an embassy in Pyongyang, has offered consular services for U.S. citizens, including those who had been detained in the North on charges of illegally entering the country or engaging in espionage acts.
The North Korean mission at the United Nations in New York has also provided a back-channel negotiation option for the two countries.
It's not the first time the countries have floated the idea of a liaison office as way to improve ties.
In 1994, when the U.S. and North Korea signed a now-defunct landmark disarmament-for-aid deal, they vowed to open liaison offices and reduce barriers to investment and ultimately have ambassador-level relations. But the idea was shelved after a nuclear crisis erupted again in 2002, with U.S. officials accusing North Korea of running a covert uranium-enrichment program in violation of the 1994 deal.
The U.S. could look to South Korea as an example. North Korea and South Korea opened an inter-Korean liaison office last September at the North's border town of Kaesong as part of a flurry of reconciliation steps.
It is the first such Korean office since the peninsula was split into a U.S.-backed, capitalistic South and a Soviet-supported, socialist North in 1945. The Koreas had previously used telephone and fax-like communication channels that were often shut down in times of high tension.
South Korean officials work at the office and regularly hold meetings with North Korean officials. They sleep at nearby lodgings and take turns staffing the office on weekends.
The town is where the Korea's now-stalled jointly run factory complex was located. It combined South Korean initiatives, capital and technology with North Korea's cheap labor. Both Koreas want the U.S. to allow sanctions exemptions to allow the reopening of the factory park, which provided the North with much-needed foreign currency.
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