"This, in no way, represents what Burnley Football Club stands for and we will work fully with the authorities to identify those responsible and issue lifetime bans.
"The club has a proud record of working with all genders, religions and faiths through its award-winning Community scheme, and stands against racism of any kind.
"We are fully behind the Premier League's Black Lives Matter initiative and, in line with all other Premier League games undertaken since Project Restart, our players and football staff willingly took the knee at kick-off at Manchester City.
"We apologise unreservedly to the Premier League, to Manchester City and to all those helping to promote Black Lives Matter."
Since the Premier League resumed last week after a 100-day hiatus due to the coronavirus pandemic, players and officials have been showing their support for the worldwide protests sparked by the killing of George Floyd by police in the United States.
Floyd, a 46-year-old unarmed black man, died as a white police officer knelt on his neck for nearly nine minutes. His death has sparked a mass movement for racial equality.
Fare, the anti-discrimination network for European Football, said the stunt was part of a winder "racist backlash".
"The racist backlash against the Black Lives Matter movement across Europe is a trend we have seen and documented," Fare's executive director Piara Powar told PA.
"Set against the BLM message of equal rights 'White Lives Matter' can only be motivated by racism and a denial of equal rights. It shows exactly why the fight for equality is so important and why the majority of people have supported it.
"At this stage we don't know who is behind this, but it's clear they don't see the relevance of the message to football or the impact that racism has on so many peoples lives.
"So be it. The movement, the issues that are being discussed and the change that will arise is unstoppable. History will judge that this was a moment that led to change."
Many of the league's top stars have also spoken out in support of the Black Lives Matter movement, including City and England forward Raheem Sterling, who said it was a "massive step" that players took a knee during the opening night of the league's restart.