Speaking in Hanoi after his summit with Mr Kim broke down, Mr Trump said he did not believe the North Korean leader was aware of Otto Warmbier's ordeal.
Mr Trump said: "He tells me he didn't know about it, and I will take him at his word."
Warmbier was jailed in North Korea in December 2015 during an organised tour.
Pyongyang authorities returned the 22-year-old to the US in a coma in July 2017, and he died days later in his hometown of Cincinnati, Ohio.
His case raised tensions at a time when the Trump administration was exchanging vitriolic comments with North Korea.
Mr Trump told reporters he had mentioned the Warmbier case to Mr Kim and the North Korean leader said he was not aware of it at the time, and was regretful.
Media captionSpeaking after his arrest, US student Otto Warmbier said he was "a poor scapegoat"
The US president said: "I did speak about it, and I don't believe that he would have allowed that to happen.
"It just wasn't to his advantage to allow that to happen. Prisons are rough, they're rough places, and bad things happened."
He said Mr Kim "felt very badly about it", adding: "He knew the case very well, but he knew it later. In those prisons, those camps, you have a lot of people."
Last June, at Mr Trump's first summit with North Korea's leader, he credited Warmbier's death with the change in US-North Korea relations.
US President Donald Trump (R) holds a meeting with North Korea"s leader Kim Jong Un during the second US-North Korea summit at the Sofitel Legend Metropole hotel in Hanoi on February 28, 2019Image copyrightAFP
Image captionMr Trump described the leaders' relationship as "very strong"
"I think without Otto, this would not have happened," Mr Trump told reporters.
"Something happened from that day. It was a terrible thing. It was brutal. A lot of people started to focus on what was going on, including North Korea. I really think that Otto is someone who did not die in vain."
Warmbier's parents were guests of honour at President Trump's State of the Union address in 2018.
The family have said their son was "murdered" by the North Korean authorities.
American student Otto Warmbier"s parents Fred and Cindy Warmbier cry as U.S. President Donald Trump talks about the death of their son after his arrest in North Korea during the State of the Union addressImage copyrightREUTERS
Image captionWarmbier's parents at Mr Trump's State of the Union address
Warmbier, a University of Virginia student, was detained in Pyongyang during an organised tour, accused of stealing a hotel poster.
He was later charged with spying for the US and held for 17 months.
This is not the first time Mr Trump has expressed a willingness to believe a foreign leader.
His response to Mr Kim echoes his reaction to Russian President Vladimir Putin regarding Russian interference in the 2016 US election.
"Every time he sees me, he says, 'I didn't do that,'" Mr Trump said after meeting Mr Putin in Asia last year. "I really believe that when he tells me that, he means it."
Amid backlash from lawmakers and US media over appearing to believe a foreign power over his own intelligence officials, Mr Trump later clarified his statements, saying he accepted the intelligence community's conclusion, but adding: "Could be other people also. A lot of people out there."
More recently, with the murder of journalist Jamal Khashoggi, Mr Trump continued to defend Saudi Crown Prince Mohammed bin Salman despite CIA officials believing the prince ordered the killing.
"[It] could very well be that the crown prince had knowledge of this tragic event - maybe he did and maybe he didn't!"
"I hate the cover-up," Mr Trump told reporters. "And I will tell you this, the crown prince hates it more than I do. And they vehemently deny it."

What's the reaction?

The Democratic leader of the House of Representatives, Speaker Nancy Pelosi, pointed out that Mr Trump had believed Mr Putin in the past.
The California lawmaker said: "It's strange. I don't know. There is something wrong with Putin, Kim Jong-un - in my view, thugs - that the president chooses to believe."
House Intelligence Committee chairman Adam Schiff, another California Democrat and a fierce Trump critic, said the president's response was "detestable".
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Even some of the president's fellow Republicans voiced muted criticism.
Senator Rob Portman of Ohio, who was with Warmbier when he returned home in 2017, said "we should never let North Korea off the hook" for what happened to the young man, the Cincinnati Enquirer reported.
Ohio congressman Brad Wenstrup, said Warmbier's "imprisonment and death were heinous crimes at the hands of the brutal Kim Jong Un regime".
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Former United Nations ambassador Nikki Haley did not rebuke the president outright, but tweeted: "Americans know the cruelty that was placed on Otto Warmbier by the North Korean regime."
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Former Republican Senator Rick Santorum told CNN it was "reprehensible" for Mr Trump to give "cover...to a leader who knew very well what was going on with Otto Warmbier".
Right-wing commentator Ben Shapiro called Mr Trump "pathetic".