Toure's current deal runs until the summer of 2015, but under FIFA rules he is able to buy out the remainder of his contract and seek a new club from this summer onwards.
Dimitri Seluk, the 29-year-old's representative, claimed City has until the weekend to offer his client an extension, or he will look elsewhere at the end of the season.
“If he signs a contract in the next three or four days, okay," Seluk told the Sun. "If not, we will not wait any longer and will start negotiations with different clubs. Today is Tuesday, so it means by Saturday. We will say 'Thank you. Okay, Yaya will leave in May.'
“I do not feel he will stay at City. This is not about money. He wants to leave because of many reasons. He is one of the best players at Manchester City but does not feel the respect of the staff. Only Roberto Mancini respects Yaya.”
The 29-year-old is already one of the top earners at City, but would become a huge target on the transfer market is he was to leave City.