The 19 year-old centre-half has publicly declared he wants ASSE to sell him to the Foxes this summer.
After seeing a previous offer rejected, Leicester have now returned with an improved bid, says L'Equipe.
The Foxes have offered a flat €32m for Fofana, including 10 per cent of any sell-on fee. Their former bid was for €25m plus €3.5m in bonuses.
The new bid on the table would make Fofana the richest transfer in St Etienne history.