Kante limped off in the first half of France's 1-0 World Cup qualifying win over Bulgaria last Saturday, ruling him out of Tuesday's 2-1 victory over Belarus.
Having had the injury assessed, boss Antonio Conte is resigned to being without him for the rest of October, starting with Saturday's trip to Crystal Palace.
"Kante is going to do another scan next week to check if his situation is improving. From the previous scan, maybe he will be out for around 20 or 21 days," Conte said.
"In Kante's situation you have to have another scan to understand the situation very well. He is a big loss for us because we know very well the importance of Kante to our team and we don't have another player with his characteristics."
However, there is some positive news on Alvaro Morata, who could return from his hamstring injury in time for the Champions League game against Roma.
The forward limped off in the 1-0 defeat to Manchester City but Conte says he is making good progress in his recovery.
He added: "I hope to have Morata back very soon, I hope for the next game."