"We want to denuke the entire peninsula", Trump said.
The U.S. insists this is the start of a process but the rather vague promises has some concerned that the mistakes of the past could be made again.
Trump and Kim met in Singapore on Tuesday for their countries' first-ever summit talks. "That's not part of the equation right now".
"Wide-ranging and profound views on the issue of establishing new DPRK-US relations, the issue of building a permanent and durable peace-keeping mechanism on the Korean Peninsula, the issue of realising the denuclearisation of the Korean peninsula and other issues of mutual concern, as required by the changed era, will be exchanged at the DPRK-US summit talks", KCNA reported in English. "That is we're prepared to take actions that will provide them sufficient certainty that they can be comfortable that denuclearization isn't something that ends badly for them, indeed, just the opposite, that it leads to a brighter better future for the North Korean people".
He called the drills "provocative", and said ending the exercises will also save the USA money.
Exercises with South Korea cover a range of eventualities, including drills with nuclear-capable bombers.
If the showy drills do end, it would be a concession on the part of the president to Kim.
"The US armed forces will in no way stop individual and collective training, the South Korean military will continue to train and prepare as they always have", he told AFP, adding that institutional knowledge from years of drills would be retained.
In a later interview with the Voice of America, Trump said the North Koreas were "very happy" about his decision to freeze the exercises "because it is so provocative".
"There were no surprises", she said.
"What is most troubling about all this is that the US gave up something tangible, namely, US-RoK military exercises, in exchange", he added, using the abbreviation for South Korea's official name.
Mr. Trump announced that the United States was withdrawing from the Iran nuclear deal last month, defying last-ditch diplomatic efforts by his European allies to convince him otherwise.
But Trump refused to endorse a communique at the G7 summit over the weekend that said participants were "guided by our shared values of freedom, democracy, the rule of law and respect for human rights and our commitment to promote a rules-based worldwide order".