The 29-year-old midfielder has reportedly threatened the reigning Premier League champions to buy out his own contract if he isn't offered improved terms by the weekend.
Under FIFA rules, players over the age of 28 are allowed to buy out their contracts, if they have not been offered a new deal after three years at a club.
Toure's apparent unhappiness at the Etihad Stadium has alerted potential suitors, with Chelsea owner Roman Abramovich seeing the powerful Ivory Coast international as the perfect replacement for Frank Lampard, who is set to leave Stamford Bridge at the end of the season.
However, the Blues are likely to face serious competition for Toure's signature, as Real Madrid also consider a move to bring the former Barcelona midfielder to the Santiago Bernabeu.
Jose Mourinho is a known admirer of the African enforcer, who still has two years to run on his £220,000-a-week deal at Manchester City.
If Toure does buy out his contract, both Chelsea and Real Madrid might be prepared to pay the player the equivalent in a signing on fee, which would mean no need to deal with the club directly.
Citizens boss Roberto Mancini will be determined to keep Toure, who was a star performer for Manchester City during their title-winning campaign last year, and hopes that an agreement can be reached between player and club.