Hundreds of fans gathered outside the club’s ground following Sunday’s 2-0 Betfred Cup defeat to Ross County, the latest poor result in a run that has seen Celtic win just two of their last ten matches.
Supporters called for the departure of Neil Lennon and players also came in for abuse from some fans as they left in convoy in their vehicles amid a heavy police presence.
A Celtic statement read: "Everyone at the club is hugely disappointed by this afternoon's result, one which has ended our phenomenal run of 35 consecutive cup victories.
"While we sincerely share the huge disappointment of all Celtic supporters, there can be no excuse for some of the violent scenes at Celtic Park this evening. The club will be investigating these events fully.
— Celtic Football Club (@CelticFC) November 29, 2020
"For players and a management team, who have given so much in recent years and have delivered 11 consecutive trophies, to require an escort from Celtic Park while being targeted with missiles, is simply unacceptable.
"While we understand that only a small number of people were involved in this behaviour, some of the actions this evening, which have obviously left our own players shaken, cannot be condoned in any way."
Chants of "Lennon, Lennon, get to f***" were clearly audible as the Celtic manager walked into his post-match media conference as the pressure mounted on the former Hoops skipper following a run of two wins in 10 matches.
A police helicopter circled overhead and more than a dozen police vans lined up outside the main stand with a number of other police vehicles parked nearby.
Footage on social media showed some fans kicking down temporary barriers that were quickly erected to separate them from those leaving the stadium, and police said two of their officers had been injured in the disturbances.
Superintendent Stevie Dolan said: "Around 4.30pm on Sunday, 29 November, a large number of fans started to gather outside Celtic Park to protest.
"An appropriate policing response was carried out and the group has now dispersed.
"No arrests have been made, however two officers sustained minor injuries as a result of the actions of the gathered group.
"We strongly condemn these actions and remind fans that enforcement options remain at our disposal.
"The Scottish Government regulations are clear that protests are prohibited in areas under level four restrictions and we would urge people to find alternative ways to protest to prevent the spread of coronavirus.
"Disorder of any sort will not be tolerated and appropriate action will be taken where any offences are identified."
Additional reporting by PA.