The live finale of Destination Mir will feature the winner boarding a Soyuz rocket and blasting into space.
NBC will continue coverage of the winner's voyage throughout the 10-day visit onboard Mir and the eventual return to Earth.
The contestants will be trained at Star City, Russia, and each week, one will be eliminated from training and will be sent home by Russian space officials.
Burnett will also get a cut of advertising revenue from the show.
The Journal reports applications for the show will likely be accepted over the Internet later this year.
NASA passed on the project, the Journal reports.