The Canaries are expected to complete a two-year deal for the German this week, according to Press Association Sport.
Farke will become Norwich's first foreign manager after sporting director Stuart Webber targeted him to replace Alex Neil, who was sacked in March.
Talks have been ongoing for weeks, with Webber believed to have met Farke in Germany, while Norwich considered several English options, including Barnsley's Paul Heckingbottom and Lincoln's Danny Cowley.
Farke has been in charge of Dortmund's second side since David Wagner joined Huddersfield in 2015 and guided the team to second in their division with only three defeats this season.