The German international is out of contract at the end of the season and as things stand he has not penned a new one and could leave on a free transfer in July or a cut-price deal in January.
Liverpool are keen to avoid both situations and want him to sign a new deal as soon as possible, but news of City's interest could make that job a whole lot harder.
Reports claim that Can would be tempted by the opportunity to work with City boss Pep Guardiola, who has high hopes of winning major honours over the coming seasons, and wants the midfielder to be a part of it.
Can is seen as the perfect replacement for Yaya Toure, who is coming towards the end of his time at the Etihad Stadium, and City could launch a move as soon as the transfer window opens again in the new year.
City's chances of concluding a deal have also been boosted by the news long-term admirers Juventus are no longer interested and have pulled out of the race.