Antonio Conte wants to bring at least three more new players into his Chelsea team before the close of the summer transfer window, according to reports in the British media.
Leicester have already rejected one offer from Chelsea of around £15million for Drinkwater this summer.
It is understood that Drinkwater is interested in a move to Chelsea, even though he may find himself behind N'Golo Kante, Tiemoue Bakayoko and Cesc Fabregas in head coach Antonio Conte's pecking order.
Leicester City manager Craig Shakespeare is confident of retaining all the players he wants to keep in his squad but is wary of approaches from rivals as the transfer window enters its final weeks.
Drinkwater would be tempted by the opportunity to join Chelsea, who will return to the Champions League this season, although there will be no guarantees of a starting place at Stamford Bridge. Such a strong interest in Danny Drinkwater comes as a result of Nemanja Matic's sale to Manchester United. The Blues were thwarted with an initial bid for the 27-year-old, with Chelsea keeping an interest in the midfield general since last season and are in dire need for cover owing to a bevy of suspensions and injuries.
The newspaper says Everton midfielder Ross Barkley and Arsenal's Alex Oxlade-Chamberlain remain targets while they are also eyeing a bid for Virgil van Dijk if Southampton indicate they are willing to cash-in on the centre-back.