According to Standard Sports, the England midfielder, is expected to undergo medical with the Premier League champions on Monday.
Drinkwater is absent from Leicester’s squad to face Borussia Mönchengladbach this weekend, as he has traveled to London to finalize terms.
Blues boss Antonio Conte, sees the 27-year-old as a possible replacement for Nemanja Matic, who has joined Manchester United in a £40m deal.
Conte has already brought in Tiemoue Bakayoko from Monaco, but the Italian wants to sign another midfielder.
Drinkwater was an integral part of the Leicester team, that won the title in 2015/2016 and his performances earned him a call-up to the England squad in March 2016.
Last summer, N’Golo Kante, who partnered Drinkwater in that title-winning Leicester midfield, joined Chelsea for £30m and has won the league title last season.