The Uefa executive committee said in a statement it has warned both countries that on top of any sanctions over trouble in Marseille this weekend, “it will not hesitate to impose additional sanctions on the Football Association (FA) and the Russian Football Union (RFU), including the potential disqualification of their respective teams from the tournament, should such violence occur again.”
Uefa had earlier launched disciplinary proceedings against Russia in response to their fans storming their English counterparts in the stadium.
“We urge both the Football Association and the Russian Football Union to appeal to their supporters to behave in a responsible and respectful manner,” Uefa said.
Europe's football governing body said Russia had been charged with crowd disturbances, racist behaviour and letting off fireworks.
It said sanctions will be decided on Tuesday once the Russia Football Union has made its case.
Uefa said there were “segregation issues” at Stade Velodrome and said it would take action to strengthen the deployment of security personnel at stadiums.
As the players left the pitch at the end of the 1-1 draw, Russian fans were seen charging into a section occupied by English supporters and fists were thrown.
The violence though was restrained compared to the scenes before the match when hundreds of English and Russian fans fought pitched battles in the Vieux-Port area of Marseille.
An Englishman was left fighting for his life and more than 30 people were injured.
Britain's government said on Sunday it was “deeply concerned” by violence at the European Championships in Marseille and offered to send more police to France.
“The UK government is deeply concerned by the violence in Marseille last night, including reports that in some cases England fans were attacked by rival supporters,” a spokesman said in a statement issued by Downing Street.
“We have offered to send further UK police to France ahead of the next England game”.
“Uefa expresses its utter disgust for the violent clashes that occurred in the city centre of Marseille, and its serious concern for the incidents at the end of the match inside Stade Velodrome,” it said in a statement.
“This kind of behaviour is totally unacceptable and has no place in football.
“Uefa acknowledges that there were segregation issues at Stade Velodrome and will implement corrective measures to strengthen the deployment of security personnel at stadiums, in close collaboration with local authorities.”
Russia's Sports Minister Vitaly Mutko on Sunday criticised the behaviour of his country's fans following the disturbances and conceded Uefa was likely to impose a fine, but he also laid the blame on the match organisers for failing to separate supporters.