Drinkwater has yet to play for Chelsea after he was an unused substitute against his former club last weekend before missing the opening Champions League group game against Qarabag due to a calf injury.
However, Kante is looking forward to seeing Drinkwater line up for Chelsea soon and thinks that he is a good signing for the Premier League champions.
"I am pleased for him to come into Chelsea," Kante said after the 6-0 win over Qarabag at Stamford Bridge. "We need everyone, especially after our time together in Leicester. 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 and I am happy that he chose to join Chelsea and I am happy to play with him again."
Kante's form for Leicester earned him a call-up to the France squad for Euro 2016 and a £30 million move to Chelsea, where he was named the PFA Player of the Year and won his second consecutive Premier League title
In choosing to move from Leicester to Chelsea, Kante missed out on Champions League football last season as the Blues faced a year out of Europe after finishing a disappointing tenth.
The 26-year-old is happy to have finally made his debut in the competition, adding: "Special, special to play my first game [in the Champions League].
"It was something that I used to watch when I was young and to play now with Chelsea here at Stamford Bridge is special.
"I am happy to make my debut in the Champions League. I remember when I was asked a year ago [about leaving Leicester to miss the Champions League], I was sure to join this good club. We did well last season and now we have to play in the Champions League, we will fight to do well in all the competitions."
Chelsea face Arsenal at home next in the Premier League and Kante admits that his international team-mate, Alexandre Lacazette, is a man to watch out for.
"I think maybe in England, you don’t know him very well but he has started to score," Kante said.
"In France, he was a very good goalscorer who scored for many, many years and he will be a threat for Sunday’s game. We have to be ready to stop him but the most important thing is the team and we need to win."