Dortmund defender Marc Bartra has been taken to hospital after he suffered superficial injuries in the incident, the club have confirmed.
The explosions occurred shortly after 6pm BST while the vehicle was leaving the L’Arrivee Hotel and Spa and heading to the stadium for the quarter-final tie, which has been postponed until 5.45pm BST on Wednesday.
A Dortmund police statement read: “According to what is currently known, the windows of the bus were (entirely or partly) smashed and one person was injured.
“It is not yet possible to say exactly what the explosion was or exactly where something exploded.”
The police said the incident happened on Wittbraeucker Strasse, around 10 kilometres (six miles) from the Dortmund stadium Signal Iduna Park.
In the aftermath of the explosions, the Dortmund team was taken back to the team hotel.
Dortmund’s CEO Hans-Joachim Watzke said: “There has been an attack with explosives on the team bus.
“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.”
Watzke said the injuries to Spanish defender Bartra are “nothing life-threatening.”
Regarding the postponement of the match, UEFA said: “This decision was made after a meeting held at the Westfalenstadion between UEFA, representatives of the two clubs and local authorities.”
With the 80,000-capacity stadium filling up just before the scheduled kick-off time, fans were told to stay in the ground until police signalled it was safe for them to leave.
And Monaco fans inside the stadium showed their solidarity with their rivals’ by chanting “Dortmund! Dortmund!”