Russia will be thrown out of Euro 2016 if their fans cause further trouble, says European governing body Uefa.
The Russians have also been fined 150,000 euros (£119,000) following violent scenes at the game against England in Marseille on Saturday.
The suspended disqualification and fine relate only to incidents that happened inside the stadium.
There were reports of minor disturbances between rival fans in Lille on Tuesday evening.
Russia play Group B rivals Slovakia in the city on Wednesday, while England fans are congregating there before Thursday's match against Wales in the nearby town of Lens.
Football Association chief executive Martin Glenn has written to members of the England Supporters Travel Club, asking them "to act in a positive and respectful way".
French authorities are increasing security before the games - 4,000 extra police and security officials will be on duty - and the sale of alcohol will be restricted in both Lille and Lens.
The BBC's James Reevell said although English and Welsh fans were singing boisterously in bars in Lille on Tuesday night, the streets of the city were calm.
Mobile phone footage posted on Twitter on Tuesday showed two sets of fans confronting each other outside a bar, with chairs strewn across the pavement on the Place de la Gare in central Lille.
However, other footage posted on social media showed fans playing football in a calm city square.
Meanwhile, groups of Russian fans are being deported from France as a result of trouble at the tournament.
A bus carrying fans was stopped in a police operation near Cannes on Tuesday.
Officials said 43 Russian supporters had been arrested and would be charged, released or expelled from the country.
Russian supporters from the bus were later transferred to a Marseille police station where the fans were expected to be held overnight.
Russia coach Leonid Slutsky, whose team still have to play Wales and Slovakia, is confident his squad will not be excluded.
"We are sure our supporters will not do the same and will not give any reasons to disqualify our team," he said.
Russia forward Artem Dzyuba added: "We're not at a street-fighting championship. Please, let's focus on football."
But he added Russian fans were not the only ones to blame for trouble.
"I don't really understand the reaction of the British media, who have this impression England supporters are like angels who just behave themselves," he said.
"You have to be objective, there is 50-50 in every conflict. I don't see that the Russians are the only ones at fault."
England were also threatened with disqualification from Euro 2016 but were not formally charged by Uefa.
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New measures, including the alcohol bans, were announced on Tuesday after Greg Dyke, chairman of the FA, expressed "serious concerns" about the security situation.
The British government has said it will send more British police officers trained in football disorder to France before England face Wales.
England manager Roy Hodgson and captain Wayne Rooney have urged fans to "stay out of trouble".
And the FA says it has contacted the families of players to give them advice after the wife of striker Jamie Vardy was caught up in the violence in Marseille.
England midfielder Adam Lallana, who has friends and family coming to the match against Wales, said: "We've just got to hope the security is there."
Meanwhile, Slovakia football officials have urged their fans to avoid Russian and England supporters in Lille and Lens.
They say fans should not react to any provocation and to immediately leave a scene of a conflict.
Russia was given a six-point deduction, suspended for three and a half years, after supporters behaved badly during Euro 2012.
Russia can appeal against the latest Uefa decision, but although sports minister Vitaly Mutko, who is also president of the Russian Football Union, called the punishment "excessive", he indicated that would not happen.
French police blamed 150 "well-trained" Russian hooligans for clashes before Saturday's 1-1 draw against England.
Six England fans were jailed on Monday for their roles in the disorder.
Over three days of disorder in Marseille, 35 people were injured - most of them England fans - and 20 people were arrested.