The Blues are looking at the Foxes star as a potential replacement for Nemanja Matic after he was sold to Manchester United.
Sky Sports journalist Kaveh Solhekol claims the Leicester man is not for sale though.
He tweeted: "Chelsea interested in signing Danny Drinkwater from Leicester City.
"He's one of midfielders they're looking at. Drinkwater is not for sale."
Antonio Conte wants to add another midfielder to his squad despite signing Tiemoue Bakayoko from Monaco.
Other targets include Arsenal's Alex Oxlade-Chamberlain and Everton's Ross Barkley.
Drinkwater signed a new four-year contract at Leicester last summer and they are under no pressure to sell one of their key players.
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Chelsea are looking to strengthen their squad as they prepare for a return to the Champions League.
So far Conte has added Bakayoko, Antonio Rudiger, Alvaro Morata and Willy Caballero to his ranks.
The Blues have also been linked with Juventus left-back Alex Sandro.
