Merseyside Police in Liverpool are investigating the incident before the game when the visitors arrived at Anfield in Liverpool.
Police say there were approximately 400 people there and that “a number of objects” were thrown on the vehicle while pyrotechnics were also lit in the area.
Pictures of the scene show an outer pane of a broken double-glazed window, but police confirmed there were no injuries.
“It is completely unacceptable and shameful behavior by a few people. We sincerely apologize to our visitors for the misery that has arisen.
“We will work with the Merseyside Police Department to establish the facts and identify those responsible.”
Authorities say they are now going through the CCTV footage of the incident and are working closely with Liverpool to determine who was involved. They urged anyone with social media footage to get in touch.
“Such incidents will not be tolerated under any circumstances.”
Real Madrid was not immediately available for CNN Sport’s request for comment.
The game itself ended in a goalless tie, meaning Real reached the Champion League semi-finals after their 3-1 win in the first leg. Madrid have faced Chelsea in their last four games to win a record-breaking 14th European Cup.
Elsewhere in the Champions League, Manchester City reached the semi-finals after beating Borussia Dortmund 4-2 overall. There will now be a place in the final against Paris Saint-Germain.
CNN’s David Close and Homero De la Fuente contributed to the coverage.