Chelsea have identified Drinkwater as a replacement for the departed Nemanja Matic but have seen two bids turned down already – the larger of which was worth £32m.
The Foxes are understood to be holding out for a £40m offer for the 27-year-old, who was integral to their remarkable Premier League title success in 2016.
Drinkwater last summer signed a contract extension with Leicester to keep him at the club until 2021, but he has now asked to leave the King Power Stadium in order to complete a dream move to Chelsea.
Any deal would see him reunited with N’Golo Kante in midfield after the pair formed an effective partnership during Leicester’s run to the title.
Leicester are understood to have lined up Paris Saint-Germain midfielder Grzegorz Krychowiak as a possible replacement for the England international, although their preference remains to keep Drinkwater.