England manager Roy Hodgson and captain Wayne Rooney have appealed directly to fans to "stay out of trouble" after repeated violence marred the start of Euro 2016.
French prosecutors said 150 Russians were behind the most serious clashes before Saturday's 1-1 draw between the two countries in Marseille.
But both teams now face expulsion from 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", but not to travel to France without tickets.
Three days of clashes between supporters led up to England's opening fixture with Russia, with Russian fans then charging a group of England fans inside the stadium at the final whistle.
So far, 10 people, including six Britons, three French and an Austrian, face an immediate trial, Marseille prosecutor Brice Robin said on Monday.
But 150 "extremely well-trained" Russian hooligans "prepared for ultra-rapid, ultra-violent action" have not been arrested.
About 35 people have been injured, four seriously, and police have deployed tear gas to disperse football fans on a number of occasions.