Chelsea have made two offers for the Foxes defender in recent weeks.
The Blues have already had bids of £60million and £65m rejected as they seek to bolster their options in central defence.
The Foxes are, however, holding out for a fee of around £80m for the Frenchman before accepting an offer.
The Guardian says Fofana, who signed a new five-year Leicester deal in March, has now asked Leicester to come to an agreement with Chelsea, according to The Guardian.
Fofana has continued to feature for Leicester at the start of this season, playing 90 minutes in their opening day 2-2 draw at home against Brentford before also playing the full match at Arsenal where he was booked during the 4-2 loss.