The visiting team made a complaint to Liverpool after alleging a fan spat at their backroom staff during the 2-2 Premier League draw at Anfield.
City boss Pep Guardiola said he was not aware of the incident at the time but was subsequently notified.
Liverpool have gathered a number of witness statements from people seated around the dugout area and are working with City staff to collect more evidence and unredacted video footage.
The club’s official sanctions process can result in a life ban, while there is also the potential for criminal charges.
A Reds spokesperson said: “Following a complaint of a serious alleged incident near the dugout area at Anfield during the second half of Sunday’s game against Manchester City, we are conducting a full investigation, working with all relevant parties including Manchester City, to gather evidence from individuals who witnessed the incident as well as analysing all available video footage.
“The evidence collected to date is substantial and, if requested to do so, will be handed to Merseyside Police given the potential for criminal charges.
“Anfield is our home and we will ensure it is a safe, enjoyable environment for everyone attending and will not allow unacceptable isolated behaviour to impact on our club’s values and principles.”
The club’s investigation is ongoing but Merseyside Police have charged four men – one local and three from County Antrim – in relation to possession and/or throwing of pyrotechnics outside the ground.