Shortly after players from both sides took a knee to show support for the Black Lives Matter movement – a feature of every Premier League game since it returned from its coronavirus hiatus last week – the aircraft was spotted in the skies above City's home ground.
In a club statement, Burnley said those responsible for the stunt were "not welcome" at Turf Moor and reaffirmed their support for Black Lives Matter.
"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 statement read.
"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."
City were cruising with a 3-0 lead at the break, Riyad Mahrez netting twice after Phil Foden opened the scoring.