IS fighters claimed Mueller, who was seized in the Syrian city of Aleppo in August 2013, was killed in a February 6 coalition air strike that buried her in rubble.
US officials have pushed back, saying the circumstances of the 26-year-old aid worker's death remain unclear.
The girl's father expressed strong reservations about the longstanding US policy of refusing to negotiate with or pay ransom to foreign groups that have kidnapped Americans, as the Mueller family broke its media silence.
"But on the other hand, any parents out there would understand that you would want anything and everything done to bring your child home. And we tried. And we asked. But they put policy in front of American citizens' lives."
Other Western countries, in contrast, are known to have paid large ransoms to free hostages.
The interview with Mueller, his wife Marsha and his son Eric was conducted yesterday. Excerpts aired today, with the full interview set for broadcast tomorrow on other NBC programs.
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