Warmbier was seized from a tour group while visiting North Korea in January 2016 and convicted on charges of trying to steal a propaganda poster and sentenced to 15 years of hard labor.
North Korea, which has denied accusations by relatives that it tortured Warmbier, has said he was provided "medical treatments and care with all sincerity."
Sen. Rob Portman, R-Ohio, said the United States doesn't owe North Korea anything.
"Otto Warmbier was mistreated by North Korea in so many ways, including his wrongful conviction and harsh sentence, and the fact that for 16 months they refused to tell his family or our country about his dire condition they caused," Portman said. "No, the United States owes them nothing. They owe the Warmbier family everything."
Parents Fred and Cindy Warmbier are from suburban Cincinnati, Ohio.
Robert Lewis, a spokesman for the law firm that filed suit against North Korea on behalf of the Warmbier family, declined comment.
Yun told CNN on Thursday that he could not discuss details of his diplomatic discussions. He said his orders from Trump were to "do whatever" he could to get Warmbier back.
Asked if it would be unusual for the U.S. to pay medical costs of detainees, Yun said: "There was some expectation the North Koreans might raise hospital costs." He said that in past instances not involving Warmbier "some money could have been handed over, yes."
U.S. Rep. Steve Stivers, R-Columbus, said that North Korean President Kim Jong-un is to blame for Warmbier’s death.
He said the report was “further evidence of a brutal dictatorship with no respect for humanity. Kim Jong-un bears the full responsibility for the death of Otto Warmbier, and that includes the medical bills that were incurred as a result of his horrific treatment.”