US President Donald Trump and South Korean President Moon Jae-in meet in New York

US President Donald Trump and his South Korean counterpart Moon Jae-in held a summit in New York on Monday to discuss plans to restart US-North Korea talks, as Seoul's spy agency said the negotiations could take place in two to three weeks.

Negotiations aimed at dismantling North Korea's nuclear and missile programs have stalled since a failed second summit between President Trump and North Korean leader Kim Jong Un in February. Pyongyang has said it is willing to restart working-level talks in late September, but no date or location have been set.

Mr Trump and Mr Moon discussed ways to achieve practical results in the US-North Korea working-level talks, Mr Moon's office said, while Mr Trump expressed his confidence that Mr Kim will fulfill commitments made during the two summits, according to a White House statement.

"There's been no nuclear testing at all," Trump told reporters as he met Moon on the sidelines of the UN General Assembly.

"And the relationships have been very good. ... We want to see if we can do something. If we can, that'll be great. And if we can't, that's fine, we'll see what happens."

Mr Moon said he hopes working-level negotiations between the United States and North Korea will be held soon to prepare for another summit, but Mr Trump said he would want to know what would result from a third summit with Mr Kim before agreeing to hold it.

"Right now, people would like to see that happen. I want to know what's going to be coming out of it. We can know a lot before the summit takes place," Mr Trump said.

In Seoul, South Korean lawmakers briefed by a spy agency on Tuesday said Pyongyang and Washington could resume working-level talks within two to three weeks.

If both sides make progress, President Trump and Kim could meet again for a summit, before which Mr Kim may visit China for a fifth summit with President Xi Jinping, the lawmakers told reporters after a briefing by the National Intelligence Service.

During their meeting, Mr Trump and Mr Moon reaffirmed their previous pledge not to use force against North Korea, and if North Korea denuclearizes, to provide a bright future for the country, a senior official at South Korea's presidential Blue House said.

However, there was no mention of a "new method" of approach on North Korea's denuclearization during their meeting, while they talked about how sanctions must be maintained, the senior official said.

Military costs

North Korea's chief nuclear negotiator, Kim Myong Gil, last week welcomed Mr Trump's suggestion for a "new method" in talks on Pyongyang's nuclear weapons programs, saying he wanted to be "optimistic" the United States would present the "right calculation method".

Last week, Mr Trump distanced himself from a suggestion by his former national security adviser, John Bolton, for a Libyan model of denuclearization for North Korea, saying it "set us back very badly". Bolton was fired this month, with Trump naming Robert O'Brien as new national security adviser.

Before his sit-down with Mr Moon, Mr Trump said the two would discuss North Korea's repeated launches of short-range missiles in recent weeks.

"We didn't have an agreement on short-range missiles. And a lot of people and a lot of countries test short-range missiles," Mr Trump added. "There's nothing spectacular about that."

The meeting came as the allies hold discussions in Seoul on renewing a deal on how to share costs for the upkeep of American soldiers in South Korea.

Mr Trump and Mr Moon talked about how a "quick conclusion" of a new military burden sharing agreement before the end of 2019 will be a way to further strengthen their alliance, the White House said in a statement.

South Korea has shouldered part of the cost of stationing what is now about 28,500 U.S. troops in the country since a 1991 pact. In March, it signed a deal with the United States to pay 1.04 trillion won ($870.94 million) this year - an increase of 8.2% on the year.

The agreement expires at the end of this year.

Mr Trump has repeatedly urged the South to contribute more to the cost. "Talks have begun to further increase payments to the United States", he wrote last month on Twitter, adding, "South Korea is a very wealthy nation".

Mr Moon detailed South Korea's contribution to the stationing of US troops in Korea, including increasing purchases of US weapons, future purchase plans, as well as a steady increase in South Korea's share of costs, the senior Blue House official said.