The England and Manchester United legend passed away on Saturday, aged 86, with clubs across the English Football League paying their respects during the latest round of fixtures.
However, a group of City fans were heard singing offensive chants about the 1966 World Cup winner during the team's 2-1 win against Brighton at the Etihad Stadium.
As BBC Sport reports, City have now issued an official statement expressing their disappointment, and have vowed to ban the supporters responsible for the chants.
"The club condemns these chants in the strongest terms and apologises unreservedly to the family and friends of Sir Bobby, and to all those at Manchester United," the statement reads.
"On this day of all days, when the stadium stood to pay tribute to our own legend in Francis Lee, Manchester City supporters should understand and appreciate as keenly as anyone the need for respect in our game.
"Our security team are studying CCTV footage of the concourse areas. We are thankful to those who have already come forward to report this matter.
"We continue to appeal for any information that can help us identify the individuals involved so that we can take the appropriate action to issue banning orders."
Charlton's former club United will play host to City in the first Manchester derby of the 2023-24 Premier League season next weekend.
City boss Pep Guardiola has confirmed his team will be taking part in a special pre-match tribute to Charlton, who scored 249 goals in 758 appearances for the Red Devils.