Fofana is Leicester’s top target and the 19-year-old defender has agreed a five-year contract with the Foxes to move to England.
The teenage defender is pushing for the move, but Saint-Etienne general manager Claude Puel is reluctant to sell him and the club have rejected three bids from the Foxes thus far.
According to French sports daily L’Equipe, Leicester made another offer for Fofana over the weekend and have improved upon their previous three bids.
The Premier League side have offered a deal worth €40m to Saint-Etienne, which includes €5m in performance-based add-ons.
And it has been claimed that Saint-Etienne have not rejected the offer from the Foxes yet.
Leicester have not heard anything from the French club since tabling the fourth offer.
Puel remains reluctant to lose Fofana but the Saint-Etienne board are grappling over the decision as the offer is big, particularly in the current climate.
Saint-Etienne are torn over whether to accept the big offer on their table or adhere to what Puel wants, which is keep Fofana for one more season.