The report claims the Foxes' offer is at £8 million, as they look at potential replacements for Esteban Cambiasso.
Kante helped promoted side Caen to 13th in Ligue 1 last season, catching the eye of a number of teams, with The Sun claiming Marseille are interested too, potentially as a replacement for Giannelli Imbula.
French website mercato365 reported earlier this month that Tottenham Hotspur were considering a move for the 24-year-old.
The Mirror reported last month Southampton were tracking him as an alternative to Morgan Schneiderlin, suggesting he could be available for as little as £5 million, while TalkSport claimed Arsenal and Newcastle United are also in the hunt.
Leicester's offer could therefore be the opening salvo in a bidding war, or by moving early they could catch their Premier League competitors cold and wrap up a deal for the highly rated defensive midfielder before pre-season begins.