The versatile 29-year-old midfielder, who can also play at left-back, would have been out of contract next summer, and has apparently opted to move to the German champions
According to SportBild, the Bundesliga champions have paid approximately €500,000 for the player who had just one year left on his contract with Die Roten Teufel.
It is believed that the versatile Kirch will act as cover for first-choice right-back Lukasz Pisczcek, who is rumoured to be on the radar of a host of top clubs on the continent.
The 29-year-old joined Kaiserslautern from Arminia Bielefeld in 2010, but has been linked with a move away from the Fritz Walter Stadion following the club's demotion to the second division.
Kirch would be the third new arrival in Dortmund, with Jurgen Klopp's side already having confirmed the capture of Marco Reus and Leonardo Bittencourt for next season.