The 25-year-old Poland international striker has long been linked with a move to the reigning German and European champions and is now set to follow former team-mate Mario Gotze to the Allianz Arena after agreeing a five-year deal.
Lewandowski joined Dortmund from Lech Poznan in 2010 and has made 115 Bundesliga appearances, scoring 65 goals.
He was part of the side that won the Bundesliga in 2010-11 and 2011-12 and featured in Dortmund's run to last season's UEFA Champions League final at Wembley, where they lost 2-1 to Bayern.
Jurgen Klopp's men have been well off the pace in the title race this season, sitting 12 points behind leaders Bayern in fourth.
Despite that poor showing, Lewandowski has netted 16 goals from 26 appearances in all competitions for Dortmund so far this term.
