The Blues spent £35million to sign Drinkwater from Leicester City last summer.
But the Englishman only played 12 Premier League games during the 2017-18 campaign.
Despite a number of English clubs expressing interest in his services, Drinkwater remained a Chelsea player at the close of the Premier League transfer window.
But he could still join a European club either on loan or permanently before the end of August.
Daily Star, UK, reports that Sarri has now asked Chelsea that Drinkwater should be sold.
Sarri already has midfield options such as £57m summer signing Jorginho, loan arrival Mateo Kovacic, N’Golo Kante, Cesc Fabregas, Ross Barkley and Ruben Loftus-Cheek.