The Daily Telegraph is reporting that the 27-year-old midfielder is keen on a move to Stamford Bridge this summer as Antonio Conte looks to further bolster his squad.
However, the same report says that Chelsea are yet to return with a second bid after failing with an opening offer of just £15m for the England international.
The same story says that Leicester City value Drinkwater at more than £30m and that the player is not currently involved in any talks.
Chelsea are believed to be on the lookout for midfield reinforcements this summer and the same article says that the west London side are keeping tabs on the likes of Ross Barkley and Alex Oxlade-Chamberlain.
The Blues have signed four players so far this summer after having tied up deals for Willy Caballero, Antonio Rudiger, Alvaro Morata and Tiemoue Bakayoko.
Conte is aiming to get his Chelsea side to defend their Premier League title after their impressive triumph last season.
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