At a news conference after the historic meeting with Kim in Singapore, Trump announced he would halt what he called "very provocative" and expensive regular military exercises that the United States holds with South Korea.
That was sure to rattle close allies South Korea and Japan. North Korea has long sought an end to the war games.
The two leaders promised in a joint statement after their meeting to work toward the "denuclearisation" of the Korean Peninsula and the United States promised its Cold War foe security guarantees, but they offered few specifics.
Critics at home said the US president had given away too much at a meeting that gave international standing to Kim. The North Korean leader is deeply isolated, his country accused by rights groups of widespread human rights abuses and under UN sanctions for its nuclear and ballistic missile programmes.
"We will be stopping the war games which will save us a tremendous amount of money, unless and until we see the future negotiation is not going along like it should. But we'll be saving a tremendous amount of money, plus I think it's very provocative," Trump said.
The presidential Blue House said it needed "to find out the precise meaning or intentions" of Trump's statement, while adding that it was willing to "explore various measures to help the talks move forward more smoothly".
Pentagon officials were not immediately able to provide any details about Trump’s remarks about suspending drills, something that the US military has long resisted.
A spokeswoman for US military forces in Korea said it had not received any direction to cease joint military drills.
"USFK has received no updated guidance on execution or cessation of training exercises - to include this fall's scheduled Ulchi Freedom Guardian," US Forces in Korea spokeswoman Lieutenant Colonel Jennifer Lovett said in a statement.
"I was shocked when he called the exercises 'provocative,' a very unlikely word to be used by a US president," the official said, speaking on condition of anonymity because it was a politically sensitive issue.
Current and former US defence officials expressed concern at the possibility that the United States would unilaterally halt military exercises without an explicit concession from North Korea that lowers the threat from Pyongyang.
The US-South Korean exercise calendar hits a high point every year with the Foal Eagle and Max Thunder drills, which both wrapped up last month.
The Republican chairman of the US Senate Foreign Relations Committee, Bob Corker, said it was difficult to assess what had happened at the summit.
"While I am glad the president and Kim Jong Un were able to meet, it is difficult to determine what of concrete nature has occurred," Corker said in a statement.
Speaking about the military exercises, Corker told Reuters: “I don’t know if that’s an agreement or an ad hoc statement that was made. It wasn’t in the agreement and sometimes things are said and walked back after talking to people at the Pentagon and other places."
US House of Representatives Speaker Paul Ryan, a Republican, called North Korea a "brutal regime" and urged Trump to continue "maximum economic pressure" as negotiations advance.
US Senate Democratic leader Chuck Schumer faulted Trump's agreement with Kim as short on details, saying the United States gave up "substantial leverage."
The two leaders smiled and shook hands at their meeting at the Capella hotel on Singapore's resort island of Sentosa, and Trump spoke in warm terms of Kim at his news conference afterward.
Just a few months ago, Kim was an international pariah accused of ordering the killing of his uncle, a half-brother and hundreds of officials suspected of disloyalty. Tens of thousands of North Koreans are imprisoned in labour camps.
Trump said he expected the denuclearisation process to start "very, very quickly" and it would be verified by "having a lot of people in North Korea".
He said Kim had announced that North Korea was destroying a major missile engine-testing site, but sanctions on North Korea would stay in place for now.
It was unclear if negotiations would lead to denuclearisation, or end with broken promises, as happened in the past, said Anthony Ruggiero, senior fellow at Washington's Foundation for Defense of Democracies think tank.
"This looks like a restatement of where we left negotiations more than 10 years ago and not a major step forward," he said.
The joint statement said Trump "committed to provide security guarantees" to North Korea and Kim "reaffirmed his firm and unwavering commitment to complete denuclearisation of the Korean Peninsula".
North Korea has long rejected unilateral nuclear disarmament, instead referring to the denuclearisation of the peninsula. That has always been interpreted as a call for the United States to remove its "nuclear umbrella" protecting South Korea and Japan.
Kim said after the summit he and Trump had "decided to leave the past behind. The world will see a major change."
Trump's meeting with Kim followed days of him berating traditional US allies such as Canada and Germany in trade disputes. He left a Group of Seven summit in Canada early last weekend, and described host Prime Minister Justin Trudeau as "very dishonest and weak."
But it did say the two sides had agreed to recover the remains of prisoners of war and soldiers missing in action, so they could be repatriated.
Daniel Russel, formerly the State Department's top Asia diplomat under the Obama administration, said the absence of any reference to the North's ballistic missiles was "glaring."
"Trading our defence of South Korea for a promise is a lopsided deal that past presidents could have made but passed on," he said.
Trump said China, North Korea's main ally, would welcome the progress he and Kim had made.
"There is no concrete detail on the denuclearisation of the Korean peninsula and the provision of security guarantees by the United States," Li said. "It is too early to call it a turning point in North Korea-US relations."
Trump said he had formed a "very special bond" with Kim and relations with North Korea would be very different in future. He called Kim "very smart" and a "very worthy, very hard negotiator".