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."
One of the Dortmund players, Mathias Ginter, was involved in the Germany-France match that was targeted in the terror attacks in Paris on 13 November, 2015. Another present in Paris, Andre Schurrle, was injured for the Monaco match.
FC Barcelona, Bartra's former club, tweeted: "All of our support to @MarcBartra, @BVB and their fans."
Spanish Prime Minister Mariano Rajoy also tweeted his support, wishing Bartra a quick recovery.
Fifa president Gianni Infantino said the football organising body condemned the incident and wished Bartra a "speedy recovery".
Uefa president Aleksander Ceferin said he was "deeply disturbed by the explosions".
"The decision taken to postpone the UEFA Champions League match between Borussia Dortmund and AS Monaco was the correct one since we must always prioritise the safety and security of all fans, team officials and players," he said.
"I would like to express my sincere gratitude to the representatives of the two clubs, the local authorities, and the supporters for their cooperation on the ground."
Dortmund are currently fourth in the German Bundesliga table.
|Windows were broken and tyres burst in the blasts. EPA|
Borussia Dortmund to 'pull together' against Monaco after explosion sees game postponed
"Shocked" Borussia Dortmund players will "pull together" when they face Monaco, 24 hours after their team bus was damaged by explosions in Germany.
Defender Marc Bartra fractured his wrist in the incident, which led to Tuesday's Champions League quarter-final being postponed.
Chief executive Hans-Joachim Watzke said: "In such a crisis situation, all of Borussia will pull together.
"The team and coaches were shocked. Now we must channel it in some way."
Dortmund said Bartra had an operation later on Tuesday after "breaking the radial bone in his arm and getting bits of debris lodged in his hand".
Captain Marcel Schmelzer said: "We're all in shock and our thoughts are with Marc. We hope that he will make a speedy recovery."
The bus was damaged at 18:15 BST on Tuesday - 90 minutes before kick-off - when it was about six miles from the Westfalenstadion in Dortmund.
Police said there were three explosions in the vicinity of the coach. They called it "a targeted attack" and found a letter at the scene, but did not disclose its contents.
Kick-off was initially delayed, before, with thousands of fans already inside, the match was finally postponed 15 minutes before the scheduled kick-off, with Monaco fans chanting in support of their opponents.
Fifa president Gianni Infantino condemned the incident, while Uefa president Aleksander Ceferin said he was "deeply disturbed" and praised the decision to postpone the game.
Police are preparing for a "large deployment" at Wednesday's match - which is scheduled to kick-off at 17:45 BST.
Watzke said: "I have to express a huge compliment to our fans, who have dealt with it very well, objectively, reasonably and solidly.
"It will not be easy to get that out of the mind. I think the team will feel it on Wednesday."
With the second leg in Monaco set for next Wednesday 19 April, Watzke said that there was no choice but to play the game on Wednesday as Monaco have a domestic game against Dijon on Saturday.
Soon after the match was rearranged, people in the Dortmund area offered to host Monaco fans who chose to stay in Germany for an extra night or two using the hashtag #bedforawayfans.