According to Daniel Cutts of The Sun, the 32-year-old's current deal will expire at the end of the season, and Fernandinho had expected to have signed a new contract by now.
Talks have been held, but no agreement has been reached, and clubs from Spain, Italy, Germany and France are all said to be interested. No specific teams are mentioned.
Per Cutts, Fernandinho's wish is to stay at the Etihad Stadium, and he's looking for a one-year rolling contract.
Fernandinho has started the 2017-18 season in excellent form, prompting Bleacher Report's Sam Tighe to rank him among the top 15 central midfielders and defensive midfielders in the Premier League after four matches.
Despite his strong form, opinions on the Brazilian are divided. Sports writer Jordan-Luke McDonald recently asked the fans whether he should be offered a new contract at all, and only half of those who responded said yes:
The former Shakhtar Donetsk man has proved a good fit for manager Pep Guardiola's intricate system, thanks to his combination of passing ability and raw strength. He's one of the team's best enforcers in midfield but can also add to the buildup play―most midfielders usually stand out in just one of those two areas.
He's not a flashy player―since his debut season in Manchester, he's never scored more than three goals in a Premier League campaign―but his role in midfield is pivotal to letting others shine in attack.
If the player is keen on staying, there's no reason to believe a deal won't eventually be agreed on. City are likely to let Yaya Toure walk when his contract expires in summer―he'll turn 35 before the end of the campaign―and the Sky Blues won't want to lose two of their best physical midfielders in the same window.
Other clubs will have the opportunity to open negotiations in January, however, and could promise the Brazil international more opportunities or higher wages to tempt him to move.
The prospect of an easier road to silverware could also be a factor―as of right now, Fernandinho has won just a single Premier League title with City.