In a news conference, the head of Dortmund police said it was a targeted attack on the team.
Player Marc Bartra underwent an operation after breaking a bone in his wrist. No other players were hurt.
The state prosecutor said a letter found near the scene was being examined as part of the investigation.
The match was postponed until 18:45 local time (16:45 GMT) on Wednesday.
Pictures from the scene showed the bus’s windows broken and tyres burst in the blasts.
Dortmund police said that “three explosive charges had detonated” at Hoechsten outside the city at about 19:00 local time.
The first indications were that this was an “attack with serious explosives”, they said.
Fans at the 80,000-capacity Signal Iduna Park were told to stay there until it was safe to leave.
The stadium was later evacuated safely and police thanked the fans for their co-operation.
“As many officers as possible” would be deployed for tomorrow’s rescheduled game, they said.
Monaco fans were praised for their chants of support for Dortmund. Social media also carried offers from Dortmund residents to Monaco fans in need of a bed for the night on #bedforawayfans.
Borussia Dortmund said in a statement: “Shortly after the departure of the Borussia Dortmund team bus from the hotel to the stadium there was an incident. The bus has been damaged in two places.”
Dortmund’s CEO Hans-Joachim Watzke said: “The whole team is in a state of shock, you can’t get pictures like that out of your head.
“I hope the team will be in a position to be able to compete tomorrow on the pitch. In a crisis situation like this, Borussia pulls together.” (Courtesy BBC)