It's believed that the third bid from the Blues was between $60m and £70m.
Some of the add-on clauses in the offer for the 21-year-old centre-back are thought to be unrealistic by the club.
The Frenchman is not for sale despite the fact that the Foxes are yet to win a game this season.
It would take a world record fee for a defender for City to consider an offer from the Blues.
The Blues have spent a lot of money this summer, and recently had a bid for Anthony Gordon rejected.
The team is in talks with Barcelona about signing Pierre-Emerick.
Brendan Rodgers said Fofana wasn't in the right frame of mind to play and left him out of the squad for Saturday's game.
The defender signed a new five-year deal at the King Power Stadium in March.
It's not clear if Fofana will return to the squad this weekend, given that he'll be in the stands for the game against the Blues.
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