England manager Roy Hodgson and captain Wayne Rooney have appealed directly to fans to "stay out of trouble" after outbreaks of violence marred the opening weekend of Euro 2016.
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.
What has happened so far?
Three days of clashes between supporters led up to England's opening fixture with Russia, and police have deployed tear gas to disperse football fans on a number of occasions. Russian supporters then charged a group of England fans inside the stadium at the final whistle.
Six England fans were jailed on Monday for the part they played in disturbances, while a total of 20 people have been arrested - including two Russians, for invading the pitch.
However, 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. England fan Andrew Bache, 50, from Portsmouth, is in a coma after being beaten around the head by Russians armed with iron bars.
Uefa has opened disciplinary proceedings against the Russian football federation, but not the English Football Association (FA).
What have England and Russia said?
FA chief executive Martin Glenn said he is treating Uefa's warning with "utmost seriousness", while Russia sports minister Vitaly Mutko said European football's governing body had "done the right thing" to start an investigation.
However, Russian MP Igor Lebedev criticised French police and called on Russian football fans to "keep up the good work", saying there was "nothing wrong with football fans fighting".
David Davies, who was FA executive director during Euro 2000, told BBC Sport: "A team has never been as close as it is today to being thrown out of a major tournament.
"In 2000, we were just looking at what went on inside the stadia. Uefa are now looking at what goes on in the streets outside too.
"They are also under huge pressure to act decisively because of the threat of terrorism that the French authorities are already facing."
What happens next?
Sanctions against Russia will be decided at a disciplinary meeting on Tuesday, Uefa said.
Uefa handed Russia a six-point deduction suspended for three and a half years for supporters' behaviour during Euro 2012 - a penalty that would have applied to any breaches during Euro 2016 qualifying.
Russia were punished three times during the tournament four years ago for fans' actions, which included setting off and throwing fireworks, displaying illicit banners and a pitch invasion.
There were also clashes between Russia and Poland fans in the build-up to their match at Euro 2012, with 120 people arrested and 10 injured in Warsaw.
The French government will ban alcohol in "sensitive areas" and the UK has offered to send more police.
Russia's next match against Slovakia will take place in Lille on Wednesday - the day before England face Wales, just 24 miles away in Lens.
Kevin Miles, of the Football Supporters' Federation, told BBC Breakfast he was "very concerned" by the prospect of a repeat incident in Lille.
"There have been no arrests, no protection of the English fans and those Russians who have been involved over the last few days will feel free to travel to Lille and probably think they'll get away with it again," Miles said.
Assistant Chief Constable Mark Roberts, the National Police Chiefs' Council lead for football policing, said England fans should "not retaliate to provocation and seek help from police if you are harassed or attacked".
Meanwhile, the FA of Wales has asked its own supporters without tickets to avoid Lens and Lille. FAW national security officer Wayne Nash said: "With the fact Russia play Slovakia the day before, I'm cognisant of what happened on the weekend."
Police from England and Wales will hold talks with the prefect of Lens and Uefa's head of security on Tuesday.
What do the fans think?
Fans attending Euro 2016 contacted Sportsday using #bbcsportsday to share their experiences.
Some have been caught up in the crowd trouble, while others have not seen any sign of it.
Nottingham Forest fan Becky Gamester-Newton, 32, from Surrey, described scenes of "organised and unprovoked hooliganism" and said supporters needed more protection from the police.
"The police have gone in with tear gas once the attack has happened, which doesn't differentiate between the perpetrators and the victims - and in fact many of the hooligans have been equipped with masks anyway so it's actually favoured them."
James Barnett wrote: "I got tear gassed twice and charged at by about 200 Russians. I'm an accountant not a thug!"
Another fan, Andrew Armstrong, added: "Very scary, seeing that Russian gang very organised, scary to a point where I feared for our safety and families with kids."
Dominic Lissaman told Sportsday: "Walked past 20 stocky Russian men with balaclavas and batons, thankfully they didn't realise I was English, scary!"
One disgruntled fan, calling himself The Palace Addiction, was not so lucky. He wrote: "This weekend: 1) Chased by Russians 2) flight home cancelled 3) gassed by the police 4) conceded in the last minute."
Ben Johnson observed: "We've met plenty of people with stories but we have managed to miss all trouble through more luck than judgement. We were sat precisely where the Russians charged after the game but luckily left about 10 seconds before. The explosion was fairly worrying though."
But many fans have been in contact to say they have not seen any of the trouble.
Daniel Drury wrote: "Been here since Saturday morning in Marseille, not seen one punch thrown. Great atmosphere and felt safe all times."
And Ste Dooney added: "We've been in Paris since Thursday. The Eiffel Tower fan park was brilliant. Fans mingling, drinking, playing footy and no trouble."