It also said both countries must make an appeal to their fans not to cause unrest.
England play rivals Wales in the northern French city of Lens on Thursday and Russian fans will also be in the area as their team face Slovakia in nearby Lille a day earlier.
in a video message posted on the English Football Association's Twitter feed, Hodgson said: "As England manager I'm obviously very concerned about the threat which is now hanging over us and the sanctions that could possibly be imposed upon the England team.