He says that "only then will there be relief from the sanctions".
The Trump administration has been adamant that USA inspectors must be on the ground, alongside scientists from other countries, to ensure North Korea is following through on its pledge to denuclearize.
Al Jazeera's Wayne Hay, reporting from Seoul, said that the meeting was an opportunity for the three countries "to show that they are in the same page" when it comes to the issue of North Korea and denuclearisation.
Sen. Rob Portman of OH said he was "skeptical but hopeful" that the negotiations would result in North Korea destroying its nuclear weapons.
North Korea has long condemned the drills as an unacceptable act of aggression and provocation.
The Russian leader praised the historic meeting between the U.S. and North Korea, adding it was "without doubt just the first step towards a full-blown settlement".
Trump said not only was a pullout not discussed, the USA "is not drawing down at all".
Bob Menendez, the New Jersey senator who is the committee's senior Democrat, criticised Trump for not notifying South Korea and Japan, the two USA allies in the region, before he announced the drills' cancellation. But that demolition took place before the summit - and is in no way comparable to the freezing of exercises, which could signal that the U.S.
The exercises in question go well beyond routine training, which apparently is unaffected by Trump's decision. The president made the comments after his meeting with Kim.More news: North America to Hold the 2026 FIFA World Cup
On Tuesday, Trump and Kim met on the Singaporean island of Sentosa.
Secretary of State Mike Pompeo, meanwhile, briefed his South Korean, Japanese and Chinese counterparts on the Singapore summit.
"This is going to further erode people's confidence in our staying power", she said.
Foreign Minister Kono said Japan welcomed diplomatic progress with North Korea.
Other remarks from Trump surrounding his historic meeting with the North Korean leader drew attention this week.
Separately, South and North Korea have said they will completely restore their military communication lines. "Kim Jong Un is pocketing immediate concessions and not making any concrete commitments".
The meeting at Tongilgak, a building on the North Korean side of Panmunjom, was the first of its kind between the two Koreas since December 2007. Mario Ritter adapted it for VOA Learning English.
"It's a mistake to see attacks on the media as separate from those things", said Jay Rosen, a New York University journalism professor.
On this, the USA may not find China on the same page.