Burnley immediately condemned the incident, timed just as both teams had finished taking a knee in support of the Black Lives Matter movement.
Following the game, which City won 5-0, Burnley captain Ben Mee said he was "embarrassed and ashamed" and now Lancashire police are looking into the incident.
"We are making enquiries to establish the full circumstances surrounding this incident and we will then be in a position to make an assessment as to whether any criminal offences have taken place," Chief Supt Russ Procter said.
"We recognise that this banner will have caused offence to many people in Lancashire and beyond and we continue to liaise closely with our partners at both the club and in the local authority."
Sources have told ESPN that some City staff were aware of the possibility of the fly-over after a trial run was carried out the day before.
Mee said the Burnley players were aware shortly before leaving the tunnel and said the incident had affected the team's performance.
"We literally heard as we were coming out. We heard some whispers it was going to happen," Mee said.
"The club tried to stop it, but I've heard it's a small number of people who've arranged this. I hope it doesn't happen again.
"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 were quick to release a statement after the plane, which took off from an airport in nearby Blackpool, was spotted and promised to ban any supporters found to be responsible.
"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."
Blackpool airport has said it has banned banner flights from its base following Monday's incident.