The 34-year-old defender, who has been without a club since the end of the last season, has been training with Dortmund for a few days.
"Manuel Friedrich is very experienced and left a good impression in training," Dortmund's director of sports Michael Zorc said.
Friedrich last played for Bayer Leverkusen and before that was in Mainz, under current Dortmund coach Juergen Klopp.
"It's an exceptional situation," Dortmund's chief executive Hans-Joachim Watzke said.
Bayern holds a four-point lead over Dortmund and Leverkusen in the Bundesliga and Dortmund could see its title hopes all but vanish if it falls another three points behind.
Saturday's match in Dortmund is a replay of the Champions League final, won by Bayern in London.
Bayern is still undefeated in the Bundesliga, while Dortmund is coming off a 2-1 loss in Wolfsburg. Subotic, Hummels' partner in central defense, tore right-knee ligaments in that game.
Hummels played only 20 minutes against England after coming on at the start of the second half. Schmelzer played the first half.
On Friday, Hummels scored Germany's goal in a 1-1 draw in a friendly in Italy.
Hummels had been relegated to the bench in the last couple of World Cup qualifying games by Germany coach Joachim Loew.
Bayern travels to Dortmund without midfielder Bastian Schweinsteiger, who is out long-term with an ankle injury. Winger Arjen Robben is likely to be available despite receiving a knock on his ankle while on international duty for the Netherlands.
The match also marks the return of midfielder Mario Goetze to Dortmund for the first time since he left his childhood team for Bayern at the start of the season.