The report suggests that the Premier League champions will look to conclude a deal with a second bid in the region of £25 million.
Chelsea have already failed with an initial £15 million bid for the player but Leicester value the midfielder closer to £30 million.
Drinkwater is said to be interested in a summer move to Stamford Bridge, with the Premier League champions short in midfield.
N’Golo Kante and Cesc Fabregas started Saturday as the club’s only fit senior midfielders, but after the Spaniard was sent off against Burnley, the Blues now face a selection headache for the meeting against local rivals Tottenham next weekend.
Various reports have stated that Antonio Conte wants to bring in at least three more players this summer, with the Leicester City man thought to be one of them.