The 26-year-old Saracens second row suffered the injury in training on Tuesday and will see a specialist on Thursday to determine its severity.
England head coach Eddie Jones said: "We are not ruling him out of the Six Nations at this stage."
Courtney Lawes and Joe Launchbury are now expected to pair up in the second row, with Maro Itoje at flanker.
Jones will name his starting XV for Twickenham on Thursday.
Defending champions England then face Wales at the Principality Stadium on 11 February, with Kruis' inclusion in that game now unclear.
France centre Yann David, 28, has also pulled out of the England match with a thigh injury and is a doubt for their second game against Scotland on 12 February.
France head coach Guy Noves now has to select between Gael Fickou, Remi Lamerat and Mathieu Bastareaud to form his centre partnership against England.
David is the latest France player to withdraw through injury, with flanker Raphael Lakafia, hooker Camille Chat, loose-head prop Eddy Ben Arous and centre Wesley Fofana all previously ruled out.
Lock George Kruis is out of England's Six Nations opener against France on Saturday with a knee ligament injury.