In rambling remarks in the White House's Oval Office in which he also sharply criticized China over trade, Trump said that as far as he knew the meeting with Kim was still on track, but that the North Korean leader was possibly being influenced by Beijing after two recent visits he made there.
Trump distanced himself from comments by his national security adviser John Bolton that North Korea angrily denounced when casting doubt on the summit, which is planned for June 12 in Singapore.
"North Korea is actually talking to us about times and everything else as though nothing happened," Trump told reporters at the start of a meeting with NATO Secretary General Jens Stoltenberg.
In a statement on Wednesday that threatened withdrawal from the summit, North Korea's first vice minister of foreign affairs, Kim Kye Gwan, derided as "absurd" Bolton's suggestion of a deal similar to that under which components of Libya's nuclear program were shipped to the United States.
Trump said the deal he was looking at would give Kim - a hereditary ruler who presides over a state widely criticized for serious human rights abuses - "protections that will be very strong."
"He would be there, he would be running his country, his country would be very rich," Trump said.
Trump stressed that North Korea would have to abandon its nuclear weapons.
"We cannot let that country have nukes. We just can't do it," he said of North Korea, which has been working on missiles capable of hitting the United States.
The United States has demanded the "complete, verifiable, and irreversible" dismantlement of North Korea's nuclear weapons program.
Trump told reporters that if the meeting with Kim happens then "it happens" and if not the United States will go on to the next step. Again he did not elaborate.
Cancellation of the summit, the first between U.S. and North Korean leaders, would deal a major blow to what could be the biggest diplomatic achievement of Trump's presidency, one his supporters have suggested would be worthy of a Nobel Peace Prize.