Drinkwater and Kante played next to each other in the midfield of the Leicester side which won the 2015-16 league season. Kante joined the Blues last summer and Drinkwater followed him to the west London club this summer.
The France international said: "I am pleased for him to come into Chelsea. I hope it could be something like Leicester or even more, I hope.
"We did a special year together in Leicester to win the league. It wasn't expected and I really enjoyed the season that I spent with him.
"I am happy that he chose to join Chelsea and I am happy to play with him again."