A plane was seen pulling a banner that read "White Lives Matter–Burnley" when it flew over Etihad Stadium before the start of the club's match against Manchester City on Monday.
The aircraft was seen minutes after players and match officials knelt on the field for around 10 seconds to show their solidarity with the Black Lives Matter movement.
Burnley released a statement in response, denouncing the offensive banner.
"Burnley Football Club strongly condemns the actions of those responsible for the aircraft and offensive banner that flew over The Etihad Stadium on Monday evening," the club said. "We wish to make it clear that those responsible are not welcome at Turf Moor.
"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. ... We apologize unreservedly to the Premier League, to Manchester City and to all those helping to promote Black Lives Matter."
Burnley's Ben Mee spoke strongly after the match, joining his club in condemning the actions.
“I’m ashamed," Mee said. "I’m embarrassed that a small number of fans have decided to put that around the stadium. ... These people need to come into the 21st century and educate themselves."
Since the Premier League's return last week, every club has taken a knee before kicking off and worn jerseys that say "Black Lives Matter" on the back in place of players' names. The league chose to show its support for the movement after the killing of George Floyd, a Black man who died in Minneapolis police custody on May 25. Floyd was stopped by officer Derek Chauvin, who pressed his knee on Floyd's neck for nearly nine minutes.
The Premier League's decision also comes after Bundesliga teams chose to support the Black Lives Matter movement. Eintracht Frankfurt has donned jerseys reading "Black Lives Matter," while Bayern Munich recently wore warmup shirts and armbands with "Black Lives Matter" on them.