The Frenchman played a key role in Leicester winning the English Premier League (EPL) last season, appearing in 37 games.
The club tried to cling on to Kante by offering him a better contract before Euro 2016 but could not stop him from penning a deal with the London club.
But Ranieri insisted the club would carry on nonetheless. "Last season we lost Esteban Cambiasso and everybody was crying," Ranieri was quoted as saying by Skysports on Sunday.
"Now we are crying because we lost Kante. Don't worry. Football in Leicester will keep going."
The defending EPL Champions have been able to hold onto their star forward Jamie Vardy who did not wish to join Arsenal but there are still concerns around winger Riyad Mahrez.
Mahrez might leave the club before the start of the new season but has travelled with his team-mates to California for an International Champions Cup clash with Paris Saint-Germain next weekend.