The England man was an instrumental part of the Foxes' title-winning campaign during the 2015/16 season as Leicester won a first ever Premier League crown.
With the departure of Nemanja Matic, manager Antonio Conte is still looking for another addition in midfield despite the arrival of Tiemoue Bakayoko and will be lodging a new £32m ($40.9m) bid for Drinkwater according to The Sun.
The Blues have already seen an initial bid of £15m ($19.2m) turned down with Leicester demanding around £40m ($51.1m) for the 27-year-old.
Chelsea are aware of Drinkwater's desire to depart the King Power Stadium and were confident a £30m ($38.3m) bid would have been sufficient.
However, with time running out in the transfer window, closing in a week's time on Thursday (31 August), Conte wants to complete a deal swiftly.
The report claims that while the £32m bid falls short of Leicester's £40m valuation, it is seen as a compromise offer to test the club's resolve with Foxes manager Craig Shakespeare recently revealing that "acceptable" bids could be made before deadline day.
"I've read the speculation the same as you but until we get the bids accepted, there's no difference for me," Shakespeare said before Leicester's 4-1 win over Sheffield United in the Carabao Cup. "We carry on with them."
"All the time we're talking about the speculation. We'll carry on as normal until I'm told otherwise. I've said that your worst fear is that an acceptable bid comes in, but if it does, we have to deal with it. Until it does, we have no decision to make.
"We have to be aware that the window goes to the end. We have to be aware of every scenario. It could work the other way that we're in for a player and it isn't accepted until three, two, one day before. We always have conversations in the recruitment department. We're always aware of the what-ifs."
Conte is also keeping tabs on Inter midfielder Antonio Candreva, though Drinkwater still remains his first choice as the Italian manager is willing to spend the £50m ($63.9m) he received from the sale of Matic.