Toure's representative Dimitri Seluk says talks have been ongoing for six months over an extension to his current deal, which runs to 2015.
However, Seluk says that things have not gone according to plan and he does not feel that Toure will stay.
"If he signs a contract in the next three or four days, OK. If not, we will not wait any longer and will start negotiations with different clubs," Seluk is quoted by The Sun.
"Today is Tuesday, so it means by Saturday. We will say 'Thank you. OK, 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."
Toure, who is currently on international duty, is believed to be City's highest earner already - but it is now claimed that if the new deal is not thrashed out, he will consider buying himself out of the remainder of his deal, which in theory he can do under Article 17 of FIFA's transfer regulations.
"It's all strange. It's not normal to have negotiations for six months about a contract," Seluk continued.
"They have been talking about small details all this time."
Seluk believes that City's owner Sheikh Mansour may need to intervene.
"Maybe if the owner speaks to Yaya, maybe he doesn't know the details," he said.