The tune was being sung to the Reds’ anthem ‘Allez, Allez, Allez’, with the song referring to Liverpool fans “crying in the stands” and being “battered in the streets”.
It is understood Manchester City will face no action from the Football Association over the video, who have accepted the club’s explanation of events.
After the video emerged City released a statement insisting the song relates to last season’s Champions League final in Kiev, which Liverpool lost to Real Madrid.
There are claims, which City deny, that the song relates to Liverpool fan Sean Cox, who was left in a coma by a brutal hooligan attack before last season’s Champions League semi-final against Roma.
City insist the song is not about that or the Hillsborough disaster which took the lives of 96 Liverpool fans in 1989.
A City spokesman said: “The song in question, which has been a regular chant during the 2018-19 season, refers to the 2018 UEFA Champions League final in Kiev.
“Any suggestion that the lyrics relate to Sean Cox or the Hillsborough tragedy is entirely without foundation.”
However, the video has drawn a lot of criticism, with the victim’s brother Martin Cox telling TalkSPORT he was “disgusted” with what he had seen.
“Singing and chanting those words, it’s like it gives the impression that it’s OK for people to carry out attacks like that on people in the streets,” he said.
“It’s a matter very close to our hearts and whether they say it’s about Sean or not, the first thing I think about is Sean being attacked.
“It wasn’t long ago that a Manchester City fan was attacked in Germany.
“So why they think it’s OK to sing songs about people being attacked on the streets when one of their own was attacked not long ago is very naive.
“I think they’ve tarnished themselves now by coming out with songs like that because at the end of the day they are professional footballers and they’ve only let themselves down and their club down.”
City’s season is not yet over as they will now face Watford in the FA Cup final on Saturday.
If City win it will be the first time a team has won the domestic treble – with the Citizens having already won the Carabao Cup and the Premier League.