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 euro (£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.
Meanwhile, groups of Russian fans are also being deported from France as a result of trouble at the tournament.
A French police operation to arrest 40 suspected Russian hooligans is under way.
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 forwards Artem Dzyuba added: “We’re not at a streetfighting championship. Please, let’s focus on football.”
England were also threatened with disqualification from Euro 2016 but were not formally charged by Uefa.
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There is a fear of further trouble between England and Russia fans this week given that both are likely to gather in Lille.
Russia play Group B rivals Slovakia in the city on Wednesday, while England fans are expected to congregate there before Thursday’s match against Wales in the nearby town of Lens.
French authorities have already pledged to increase security in Lens by deploying 2,400 police, gendarmes, security guards and a riot squad.
But Greg Dyke, chairman of the Football Association (FA), has expressed “serious concerns” about security in neighbouring Lille.
Drinking alcohol in the streets has been banned in Lens.
The British government has said it will send more British police officers trained in football disorder to France before the match between England and Wales.
England manager Roy Hodgson and captain Wayne Rooney have also urged fans to “stay out of trouble”.
The FA also 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 Slovakia fans should not to 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 on Saturday.
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 a total of 20 people were arrested.
