Following the departure of Nemanja Matic to Manchester United, manager Antonio Conte was still eyeing another addition in midfield despite the arrival of Tiemoue Bakayoko.
The Italian had targeted Drinkwater and recently submitted a second bid worth £32m ($41.3m), however, it was still not close to Leicester's £40m ($51.6m) valuation as the Foxes reportedly rejected it as well.
However, according to Sky Sports sources, the 27-year-old has now handed in a transfer request in an effort to expedite a move to Stamford Bridge.
Drinkwater was an instrumental part of the Foxes' title-winning campaign during the 2015/16 season as Leicester won its first ever Premier League crown.
Moving to Chelsea could also see him link up with former midfield partner N'Golo Kante again and while that might hinder new summer signing Bakayoko's chances of game time, the former Monaco midfielder has welcomed his arrival.
"Competition only makes you better. I am not scared of anyone," he told SFR Sport, as translated by Sky. "I think that there are not enough of us in midfield, especially with all the competitions we will be playing in."
"So we need another midfielder to reinforce the team, so it is a very good thing that he could be coming."
Meanwhile, Leicester are said to be targeting Paris Saint-Germain midfielder Grzegorz Krychowiak as a potential replacement for Drinkwater.
Coincidentally, Krychowiak was seen as an alternative to the England man for Chelsea with the Blues looking to bring him in on loan, but it appears now that the King Power Stadium could be a potential destination for the Polish midfielder.
Krychowiak was one of Europe's most sought-after players after his exploits with La Liga side Sevilla.
However, after moving to PSG last summer, the 27-year-old has struggled to maintain his place in the starting eleven and could become a casualty of Neymar's world record move to the club.