According to a report in Wirtschafts Woche (Business Weekly), the company will discuss the future of the factory at an upcoming board meeting.
The German publication believes that the former chief of Porsche, and now head of the entire Volkswagen Group, is keen on closing the factory in 2016. Phaeton production will be moved to a plant that builds closely related Audi or Porsche vehicles, with the transparent factory's 500 or so workers moved to other locations.
Any move to close the Dresden factory will be opposed by Volkswagen's union representatives, who under German law hold half the seats on the automaker's board. Opposition to the closure is also likely to come from residents of the Dresden, which is located in what used to East Germany.
The factory was opened in 2002 and features expansive glass walls, which allows customers to view their Phaetons during the final phases of the build process.
Although the building is considered an architectural success, the vehicle it has been assigned to build has been anything but. The current-generation Phaeton is widely regarded as the brain child of former Volkswagen chairman Ferdinand Piech.
In 2013, automotive analyst Sanford Bernstein compiled a list of the 10 biggest loss-making vehicles in Europe. According to Bernstein, Volkswagen has lost around US$2.9 billion ($4 billion) on the Phaeton.
Wirtschafts Woche believes that, at present, Volkswagen is only building eight Phaetons a week at the plant.
Earlier this year, the company confirmed that the next-generation Phaeton won't arrive until around 2020, and will be an all-electric sedan built atop a platform shared with Porsche and Audi.