The England international's future at the Etihad has been called into question over the past few months, and he is yet to pen an extension on his existing deal, which expires in 2023.
City are thought to be keen to tie Sterling down to a new deal, but the Premier League champions could be under pressure to sell the 26-year-old should his contract situation remain unchanged.
According to Sport, Barcelona failed to secure Sterling's signature on a temporary basis this summer but are prepared to reignite their pursuit of the former Liverpool man in January.
The arrival of Jack Grealish has seen Sterling struggle for minutes in the final third this season, and he has only started one of City's four Premier League games so far in the 2021-22 campaign.
The attacker boasts 115 goals and 87 assists from 297 appearances for the club in total and netted three times en route to the Euro 2020 final with England. body check tags ::