Having sold Nemanja Matic to Manchester United for £40 million, Antonio Conte is keen to bolster his midfield ranks before the end of the transfer window.
The sales of both Matic and Nathaniel Chalobah have left the Premier League champions short of options in the middle of the park, despite the arrival of Tiemoue Bakayoko from Monaco.
The former Italy boss has also looked into the possibility of bringing in Bayern Munich''s Renato Sanches, but ambitious Serie A outfit AC Milan appear to be his preferred destination.
Leicester, meanwhile, are reluctant to sell Drinkwater - 12 months on from offloading N'Golo Kante to Stamford Bridge - and reports suggest that the King Power Stadium outfit have placed a £30m price tag on the former Manchester United trainee.
The Foxes already look set to lose Riyad Mahrez, with Roma showing a serious interest in the Algeria winger, and they do not want too many of their star names to depart as they seek to re-establish themselves at the top of the Premier League table.
The arrival of Sevilla captain Vicente Iborra suggests a ready-made replacement for Drinkwater is already on board, but Craig Shakespeare will fight to keep hold of the 27-year-old.
Drinkwater played a key role as Leicester surprisingly won the Premier League title in 2015-16, but struggled for form last season as he contributed just one goal and three assists in all competitions.