A Football Association official asked the England supporters' band to stop playing during the win over Scotland after it inadvertently provided background music to anti-IRA singing.
As the band played "Follow England Away" - a song it regularly performs - some of the 5,000 away fans began the anti-IRA chant.
Manager Roy Hodgson said: "I didn't have a clue what they were chanting.
"If anyone was offended, I'm sure the FA would like to apologise to them."
An FA official contacted the band during the first half to ask it to stop playing the song because it was providing the tune for the offensive singing.
England band leader John Hemmingham has also apologised for any offence caused but stressed the musicians were unaware of the issue.
He explained: "All the fans around us were singing "Follow England Away". It was only when a band member saw on Twitter that some people were saying we were playing anti-IRA songs that we became aware.
"Then we immediately stopped and played something else."
He also insisted it was the band who brought the issue to the attention of the FA, and not the other way round.
Hodgson added: "I don't condone it. Hopefully they will behave themselves and not get themselves into a situation where their chanting is being criticised."
Police Scotland reported no football-related disorder in the immediate aftermath of the match and said they had not received any reports of offensive chants.
England won the match 3-1 with Alex Oxlade-Chamberlain putting them ahead before Wayne Rooney scored either side of Andrew Robertson's goal for Scotland.