The midfielder has not made a Premier League appearance for 18 months and The Sun has reported that he faces a further six weeks out.
Burnley have not commented on Drinkwater’s injury, but it understood to have occurred away from the training ground.
The former England man was surplus to requirements at Chelsea, with boss Frank Lampard preferring Jorginho, Kovacic and N’Golo Kante in his position.
Drinkwater moved to Burnley on loan in the summer but has only featured in the EFL defeat to Sunderland so far this season.
He has not featured in a top-flight match since March 2018 when he started in Chelsea’s defeat to Manchester City at the Etihad Stadium.
Drinkwater, who won the title with Leicester, moved to Chelsea in 2017 but started just five Premier League games for The Blues.
Sean Dyche signed Drinkwater for Burnley to provide backup for Ashley Westwood and Jack Cork in central midfield.