South Korea said Mr Trump had also said in a phone call with its President Moon Jae-in that there would be no military action while North-South talks were going on and that a Wall Street Journal article saying he was contemplating a military strike against North Korea was "completely wrong."
"Who knows where it leads?" Mr Trump told reporters at the White House after his call with Mr Moon to discuss Tuesday's North-South talks, the first since 2015.
He said Mr Moon told him the talks went well, and added: "Hopefully it will lead to success for the world, not just for our country, but for the world. And we'll be seeing over the next number of weeks and months what happens."
At a later news conference with the visiting Norwegian prime minister, Mr Trump said the United States had problems with North Korea, but "a lot of good talks are going on right now."
"I see a lot of good energy. I like it very much ... So, hopefully, a lot of good things are going to work out."
"I think that we will have peace through strength," he added, referring to his policy of ensuring peace through a strong military.
South Korea's Presidential Blue House said both men had said the dialogue "could naturally lead to talks between the United States and North Korea for the denuclearisation of the Korean peninsula after the Pyeongchang Winter Olympics."
At Tuesday's intra-Korea talks, North Korea said it would attend the Olympics, which South Korea will host next month, while both sides agreed to resolve problems between them through dialogue and to revive military consultations to avoid accidental conflict.
However, Pyongyang said it would not discuss its nuclear weapons because they were aimed only at the United States and not its "brethren" in South Korea, or Russia or China, showing that a diplomatic breakthrough to the crisis remained far off.
While Washington has welcomed the talks as a first step toward solving the crisis over North Korea's program to develop nuclear missiles capable of reaching the United States, it has reiterated that any talks involving the United States must be aimed at North Korea's denuclearisation.
The White House said Mr Trump told Moon the United States was willing to talk to North Korea "at the appropriate time and under the right circumstances."
Mr Trump, who has swung between hurling insults and threats at North Korea to expressing a willingness to talk, said on Saturday he would be willing to speak to North Korean leader Kim Jong Un, though not without pre-conditions.
An unsourced article in the Wall Street Journal on Tuesday said U.S. officials were debating whether it was possible to mount a limited military strike against North Korea without igniting an all-out war.
The Trump administration has said it prefers a diplomatic solution, but that all options are on the table, including military ones.
U.S. officials say Mr Trump has been considering a number of military options, including a preemptive strike on a missile or nuclear facility, but officials and analysts have warned of the risks of triggering a catastrophic wider conflict.
MOON CREDITS TRUMP FOR TALKS
Earlier on Tuesday, Mr Moon made a point of crediting Mr Trump for the Korean talks and also said he himself was open to meeting with Kim at any time if conditions were right and "certain achievements are guaranteed".
"The purpose of it shouldn't be talks for the sake of talks," he told a news conference, while warning that Pyongyang would face stronger sanctions if "provocations" continued.
Mr Trump and Mr Kim have exchanged threats and insults over the past year, raising fears of war. South Korea and the United States are technically still at war with the North after the 1950-53 Korean conflict ended with a truce, not a peace treaty.
Washington and Seoul opened the way for the talks last week when they announced the postponement of joint military exercises that Pyongyang has denounced as a rehearsal for invasion.
Washington had raised concerns that a New Year overture from Kim that led to the talks could be aimed at driving a wedge between the allies, but Moon said he and Washington did not differ over how to respond to North Korean threats.
"This initial round of talks is for the improvement of relations between North and South Korea. Our task ... is to draw North Korea to talks aimed at the denuclearisation of the North," he said. "(It's) our basic stance that will never be given up."
North Korea's official Rodong Sinmun newspaper said all problems would be resolved through the efforts of the Korean people alone.
"If the North and South abandon external forces and cooperate together, we will be able to fully solve all problems to match our people's needs and our joint prosperity," it said.
In spite of the hopeful words about the potential for future talks, the U.S. intelligence assessment of North Korea's weapons programs has not altered, officials say.
U.S. officials familiar with the classified analysis say the consensus is that Mr Kim remains convinced that the United States is determined to overthrow him and that only a nuclear arsenal that threatens America can deter that.
One official said the North-South talks were likely to follow the pattern of past diplomatic efforts, in which North Korea has benefited from aid without making concessions.
Lee Woo-young, a professor at Seoul's University of North Korean Studies, said it was wise of Mr Moon to praise Mr Trump.
"By doing that, he can help the U.S. build logic for moving toward negotiations," he said.
The United States and Canada are due to host a conference of about 20 foreign ministers next week in Vancouver to discuss North Korea, without the participation of China, Pyongyang's sole major ally and biggest trade partner.
China would not attend and was resolutely opposed to it, its foreign ministry spokesman Lu Kang said.
"It will only create divisions within the international community and harm joint efforts to appropriately resolve the Korean peninsula nuclear issue," he said.
(Additional reporting by Josh Smith and Hyonhee Shin in SEOUL, Michael Martina in BEIJING and David Brunnstrom, John Walcott, Tim Ahmann and Eric Beech in WASHINGTON; Writing by David Brunnstrom and Soyoung Kim; Editing by James Dalgleish)
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)
Get the latest election news, live updates and election schedule for Lok Sabha Elections 2019 on ndtv.com/elections. Like us on Facebook or follow us on Twitter and Instagram for updates from each of the 543 parliamentary seats for the 2019 Indian general elections.