French prosecutors said 150 Russians were behind the most serious clashes before Saturday's 1-1 draw between the two sides in Marseille.
Both teams face expulsion by Uefa if there is any further violence.
"I am very concerned at the threat now hanging over us," said Hodgson.
"We have worked very hard to get here and desperately want to stay.
"We appreciate all your support at the matches, of course, but I am appealing to you to stay out of trouble and to try to make certain these threats being issued are never carried out and we will be able to attempt to do the best we can to stay in this competition by football means."
Manchester United forward Rooney asked fans to be "safe and sensible", and not to travel to France without tickets.
The direct appeal echoes manager Kevin Keegan's plea to England fans after violent scenes in Charleroi, Belgium, during Euro 2000.
Uefa have written to both the FA and the Russian FA warning them that further trouble from their fans could result in expulsion from the competition and potential other sanctions.