A club spokesman said the behaviour “of a few individuals” was “totally unacceptable and shameful”.
Hundreds of fans lined Anfield Road, very few wearing masks or observing social distancing, to greet both teams as they arrived for the Champions League quarter-final second leg with a number setting off flares and smoke cannisters.
When one of Real’s buses parked up at the Kop end of the ground after the team had disembarked, it was apparent one pane of a double-glazed window had been smashed and Liverpool staff were seen cleaning up a pile of broken glass on the ground with Merseyside Police officers in attendance.
“We condemn unequivocally the actions that led to Real Madrid’s team bus being damaged during its arrival to Anfield this evening,” said a Liverpool spokesman.
“It is totally unacceptable and shameful behaviour of a few individuals. We sincerely apologise to our visitors for any distress caused.
“We will work together with Merseyside Police to establish the facts and identify those responsible.”
Chief superintendent Zoe Thornton of Merseyside Police condemned the “disgraceful behaviour” of some fans outside the ground, which she branded “extremely disappointing and totally unacceptable”.
Confirming an investigation is underway, Thornton said in a statement issued by the force: “We can confirm an investigation is under way after damage was caused to the Real Madrid team bus this evening, as it approached Anfield Stadium ahead of Liverpool’s Champions League match with Real Madrid.
“Despite the advice to avoid attending the ground, approximately 400 people attended Anfield Road and as coaches approached, a number of items were thrown at the Real Madrid team bus.
“We are aware of images which show the outer pane of the lead coach window being damaged, the glass did not break through and there are no reported injuries.
“A number of pyrotechnics were also lit in the area, which significantly reduced visibility and could have put other people in danger.
“As with all matches, we have worked very closely with both clubs to ensure the safety of the public and the teams themselves, and the disgraceful behaviour of those involved this evening is extremely disappointing and totally unacceptable.
“Such incidents will not be tolerated under any circumstances.
“A dedicated investigation team will be reviewing CCTV footage from the area, along with our own body worn camera footage, and working closely with the club to identify those responsible.
“I would also ask anyone who was in the area and captured mobile phone video footage of the incident to please get in touch.”
It is not the first time the opposition coach has been damaged in the build-up to a match at Anfield.
Three years ago Manchester City’s bus was hit by bottles and flares, ahead of another Champions League last-eight clash.
It caused so much damage City had to call for a back-up bus to take the players home after their 3-0 defeat that night.
On that occasion Uefa fined the club 20,000 euros (£17,000) for the damage as well as 6,000 euros (£5,200) for setting off fireworks and throwing objects.