The German team confirmed one of its players has been injured following the explosions 10km from the Signal Iduna Park stadium in Dortmund, North-Rhine Westphalia in east Germany.
The player is Spanish defender and father-of-one Marc Bartra, who has been taken to hospital after sustaining injuries to his hand and arm, the team added.
His injuries are worse than initially thought, and he has had surgery to repair a broken bone in his hand after shattered glass from the bus' windows splintered – despite the glass being bullet-proof.
The bombs were buried in a hedge, which police have now said they are working on the assumption the football team was "specifically targeted".
Two hours after the initial explosion a suspicious package was found at the team's hotel, just before they were due to arrive back there.
BVB boss Hans-Joachim Watzke, said: "Defender Marc Bartra was given emergency care and then taken to the hospital.
"We must now try to process the shock.
"We have to play again tomorrow. This will be a great challenge."
Spanish media has reported two of the three bombs exploded right next to where Mr Bartra was sitting.
The team had just set off from their hotel when their bus was hit.
Mr Watzke, CEO of BVB, and football official Dr Reinhard Rauball, said: "The bomb attack was in close proximity to the hotel exit."
At 8.35pm GMT, police said they still did not know what type of explosive had been used, however they are certain there were three detonations.
Police have thanked all fans for their calm behaviour.
Live images from the Borussia Dortmund attack
Tue, April 11, 2017
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A football player has been injured after three bombs exploded near Borussia Dortmund's team bus ahead of a Champions League match against Monaco
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A view on the damaged windows of the Borussia Dortmund's team bus after it was hit by three explosions in Dortmund, Germany
The quarter-final against Monaco has been postponed until tomorrow at 6.45pm after the German team put a message on the stadium screens informing fans of the incident.
All fans' tickets will remain valid and those who cannot make it tomorrow will have the opportunity to offer their tickets for re-sale from 11pm.
The stadium was then evacuated, with all fans having left by 9pm local time (8pm GMT).
A police officer, said: "That was completely peaceful and civilised."
Three bombs exploded, involving the Borussia Dortmund bus
During a press conference Prosecutor Sandra Lücke, said: "The investigation is being carried out on suspicion this was an attempt to kill.
"A letter was found near the crime scene. We will check the authenticiy intensively."
Dortmund's Police President Gregor Lange, added: "We assume that it was a targeted attack on BVB's team.
"We decided at a very early stage not to rule out anything.
"At the present time, the concrete background of the act is not very clear.
"We are preparing for a large-scale operation tomorrow.
"The protective measures for the teams have already begun. Explosive dogs are still in use."
Marc Bartra has been taken to hospital with an injured hand
The players were escorted by police back to their hotel
In a tweet, the club said: "We will inform you as soon as there is news."
Football’s governing chiefs have now come out to condemn the horror blasts.
Aleksander Ceferin, the head of Europe’s governing body UEFA, said he was “deeply disturbed” to hear about the explosions.
But he added Borussia Dortmund and AS Monaco are right to postpone the match and prioritise the safety of the fans.
And Gianni Infantino, president of FIFA, branded the events “troubling”.
He added: “FIFA condemn the incidents in Dortmund.
“We are all awaiting further details of the incident.”
— Borussia Dortmund (@BVB) April 11, 2017
Police said they are now using a drone to see if there are any more explosive devices.
The explosions are said to have come from a bush in Wittbräuckerstraße, 10km from the stadium where they were due to kick-off at 7.45pm local time (6.45pm GMT).
Monaco fans have started singing "Dortmund, Dortmund" in response to the incident, with the tunes being "very warmly received" by Dortmund fans.
A concerned Dortmund fan looks on as he hears the news
Dortmund put a tweet on stadium screens telling fans of the incident
BVB has said Monaco supporters who need accommodation should check the hashtag #bedforawayfans as BVB fans are providing them with a bed so they can stay for the re-scheduled match tomorrow evening.
The team tweeted: "Dear supporters of @AS_Monaco_EN!
"If you need accommodation in Dortmund, please check #bedforawayfans. #bvbasm."
A statement from Dortmund Police, said: “In the run-up to the Champions League match of the BVB against the AS Monaco, there was an explosion near the BVB team teambus shortly after 7 pm.
“The event location is Wittbrauckerstrasse / Schirrmannweg in Dortmund-Hochsten.
“According to current knowledge the disks of the bus (whole or partial) were broken and one person was injured.
“It is not yet possible to say exactly what the explosion was or exactly where something exploded.”