"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," a statement released on the club's website at half-time 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.
"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."
The plane appeared shortly after players from both teams took a knee in support of the Black Lives Matter movement.
There was no indication of who organised the plane and banner.
After the match, Burnley defender Ben Mee told reporters that the players had heard rumblings that a display would be made minutes before kickoff.
"We literally heard as we were coming out. We heard some whispers it was going to happen. The club tried to stop it, but I've heard it's a small number of people who've arranged this. Hope it doesn't happen again," Mee said. "I'm ashamed and upset it's associated with the club, my club, and it's not something we want to see in this game."
Burnley manager Sean Dyche said that he knew nothing of the banner and that the club could only apologise for its appearance.
"I didn't realise what had gone on at the beginning. Heard noise. Unacceptable. We can only apologise," Dyche said after the match. "I don't know what the club can do to send a message out. The powers that be will look at the way we can play our part."
Manchester City manager Pep Guardiola, who last week said that white people should apologise for the treatment of Black people, said the banner's message missed the point of the Black Lives Matter movement.
"Of course white lives matter, but black lives matter, too. Human beings matter. Everyone, we are the same. I travel a lot around the world and live in many countries, and every one of us, we are the same," he told reporters. "Every day we have to fight not just for the situation but for all the injustice around the world."
Former England and Manchester City defender Micah Richards says he was disheartened to see the plane carrying its banner.
"It's so disheartening. After how far we've come in the last couple of weeks, it really does hurt me," Richards told Sky Sports.
"I agree, everyone should have free speech, but just at a time when things are on the up, a small fraction just want to spoil things.
"Burnley have come out and condemned it, but it just shows you -- I speak to a lot of people who say, 'all that stuff happens in America, it doesn't happen in England.'
"... We can wear t-shirts, we can do 'Black Lives Matter' protests, and slowly we're getting better. But it just shows today, that although it's a small minority, it's still happening."
All players in the 12 Premier League games since the restart of the season after the coronavirus pandemic stoppage have worn "Black Lives Matter" on their shirts in place of their names in response to the May 25 killing of George Floyd, a Black man, who died after Derek Chauvin, a white police officer, kneeled on his neck for nearly eight minutes in Minneapolis. Floyd's death has spurred demonstrations against racial injustice around the world as well as in sport.
Information from Reuters was used in this report.