Three explosions around the team bus left defender Marc Bartra in hospital, where he required an operation for a broken bone in his right wrist.
The incidents occurred as the squad left their hotel en route to the Signal Iduna Park stadium for the first leg of their quarter-final tie, which was swiftly postponed.
The local police force - Polizei Nordrhein-Westfalen Dortmund - held a joint late-night press conference along with the football club and the Dortmund public prosecutor.
Police chief Gregor Lange said, in quotes reported by Bild : "This evening at 19:15 there were three explosions in the surroundings of the BVB hotel.
"We must start from (the position this was) a targeted attack on BVB's team. Strong police forces are in operation.
"We also have explosive detection dogs in use and decided to search the scene from the air. At the present time it is not clear what the background of the act is."
Dortmund police also revealed a letter had been found, claiming responsibility for the incidents.
"The prosecutor's office #Dortmund confirms the discovery of a letter near the scene of the crime. The authenticity is currently being tested," the police force posted on Twitter .
The police earlier said in a statement the explosives "may have been hidden in a hedge near a parking lot".
Goalkeeper Roman Burki was sitting at the back of the team bus alongside Bartra, and told Swiss newspaper website blick.ch how events unfolded.
Burki said: "We left the hotel at 7:15pm and went down the street. The bus turned down the main street, and there was a giant explosion.
"I was sitting in the back row next to Marc Bartra, who was hit by fragments of the broken glass.
"After the bang, we all ducked in the bus and those who could threw themselves to the ground. We did not know had happened.
"The police were quick on the scene, and we were safe.
"We're all shocked - nobody thought of a football match in this moment."
It was announced shortly after 19:30 BST that the match had been called off, and it will instead kick off at 17:45BST today.
Meanwhile, AS Monaco said on Twitter all their supporters staying in Dortmund on Tuesday night would be reimbursed up to 80 euros (£68) by the club.