Manager Antonio Conte wants to bolster his options following the sale of Nemanja Matic to Manchester United and has drawn up a list of targets.
England international Drinkwater is on that list as well as names such as Renato Sanches at Bayern Munich, Alex Oxlade Chamberlain at Arsenal and Everton's Ross Barkley.
The 27-year old Drinkwater, who would also boost Chelsea's homegrown contingent, signed a four year contract at Leicester and is key to Craig Shakespeare's plans.
Though he would be interested in a move to Chelsea he is conscious he could end up with a peripheral role as Conte aims to pair Tiemoue Bakayoko with N'Golo Kante as his midfield engine room of choice.
New recruit Bakayoko though is doubtful for the start of the season following minor knee surgery.