The Sun is reporting that the west London side are keen to bring the 27-year-old midfielder to Stamford Bridge this summer but will not pay over the odds to secure his signature.
The same report claims that Antonio Conte first signalled his interest in signing Drinkwater a year ago but a move did not materialise.
Chelsea are now said to have reignited their interest in signing Drinkwater as they look to further bolster their squad ahead of next season.
The west London side have signed four players so far this summer, having wrapped up deals for Willy Caballero, Antonio Rudiger, Tiemoue Bakayoko and Alvaro Morata.
Chelsea are gearing up for a return to the Champions League next season after they won the title in Conte’s first campaign in charge last term.
The Blues take on Arsenal in the Community Shield at Wembley on Sunday afternoon.
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