UEFA said the penalties could include "the potential disqualification of their respective teams from the tournament."
The statement urged English and Russian football authorities "to appeal to their supporters to behave in a responsible and respectful manner."
The warning came after three days of clashes between rival fans in the southern French city of Marseille, which left one fan fighting for his life and more than 30 others injured. Ten people have been arrested.
Saturday`s Euro 2016 match, which ended in a 1-1 draw, was marred by further trouble inside the Stade Velodrome.
Fans stormed towards rivals
Eyewitnesses described how Russian fans appeared almost without resistance to breach a thin police cordon separating them from the England fans. Video footage showed Russian fans charging at England supporters, forcing them to flee, with some having to climb fences and women and children seen in a state of panic.
Earlier on Sunday, UEFA already opened a disciplinary case against Russia, warning in an separate statement that: "This kind of behavior is totally unacceptable and has no place in football." A decision on penalities is expected by Tuesday.
No action has been launched against England as the violence caused by their fans was away from the stadium before the match.
Russian Sports Minister Vitaly Mutko said a fine was the most likely penalty, R-Sport news agency reported.
"We will have a fine from UEFA, so I understand. We behaved incorrectly," Mutko added. Russian officials will also investigate what happened, he added, noting what he said were "many nuances."
The British government said it was "deeply concerned" by the violence in Marseille and offered to send more police to France.
Security upgraded
Despite fans` criticism of the French police response inside the stadium, UEFA praised the work of security forces.
But a UEFA spokesman admitted that security personnel would be increased at subsequent Euro 2016 matches to segregate rival fans inside stadiums.
The spokesman said additional officers would be deployed to higher risk matches, such as Sunday`s Turkey-Croatia fixture in Paris.
Meanwhile, authorities in the northern city of Lille are considering restricting alcohol sales and are likely to reinforce riot police numbers for games later this week involving the English and Russian sides.
Russia plays Slovakia in Lille on Wednesday afternoon. England meets Wales a day later in Lens, which is just 30 kilometers (20 miles) away from Lille.
Also on Sunday, French police said they turned away 14 suspected soccer hooligans who were trying to cross into northern France from Belgium.
More about:
