The show - known as Government in English-speaking countries - was created in 2010 by the team behind the original version of The Killing.
Friday Night Lights executive producer David Hudgins will write the US adaptation and will also executive produce alongside FNL's Jason Katims, according to Deadline.
The original ten-part Borgen focuses on political leader Birgitte Nyborg (Sidse Babett Knudsen), who faces new challenges when she scores a surprise election victory.
The Danish version will also air in the US via Link TV, while BBC Four has secured the rights to the show in the UK.
Meanwhile, Universal Television will co-produce the NBC remake alongside BBC Worldwide Productions, with Jane Tranter and Julie Gardner also serving as executive producers.
Other drama projects in development at NBC include adventure thriller Revolution from JJ Abrams and Supernatural's Eric Kripke, a modern-day update of Frankenstein and an adaptation of 1984 film Romancing the Stone.
A wrestling show produced by Dwayne 'The Rock' Johnson, a romantic drama from The West Wing's Michael Oates Palmer and a a cop show starring Ethan Hawke and Vincent D'Onofrio are also in the works.
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Watch a trailer for the original Borgen below: