UEFA has told England and Russia that they could be disqualified from Euro 2016 if fan violence continues.
The executive committee of European football's governing body warned the English Football Association (FA) and the Russian Football Union (RFS) that "it will not hesitate to impose additional sanctions."
In a statement, it said these could include "the potential disqualification of their respective teams" from the tournament in France.
The statement added: "We urge both the FA and the RFS to appeal to their supporters to behave in a responsible and respectful manner.
"We would also like to publicly voice our support for the work of the French authorities and security forces for their efforts to deliver a safe and secure tournament in challenging circumstances."
The action was in connection with crowd disturbances, racist behaviour and fireworks being set off during the game at the Stade Velodrome in Marseille.
UEFA has not so far taken any action against the FA.
A large group of Russian fans in the stands behind England's goal advanced on the neighbouring England section at full-time, throwing objects and breaking through a line of stewards as England fans fled for the exits.
In addition to the confrontations, fireworks -- which are not allowed to be taken into the grounds -- were set off inside the stadium.
UEFA said in a statement that it expressed "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."
The statement added: "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."
A fine for Russia is the most likely punishment, although they could face a possible points deduction.
"We will have a fine from UEFA, so I understand. We behaved incorrectly," Russia sports minister Vitaly Mutko said in a report by R-Sport news agency.
The trouble at the Stade Velodrome came after a third day of violence in Marseille left as many as 20 England fans injured, with several reported to have been seriously hurt.
Fist fights and bottle throwing broke out between England supporters and their Russian and French counterparts on Saturday afternoon, with police using water cannons and tear gas on rioters hours before the game started.
The FA has said it is taking UEFA's warning "with utmost seriousness."
FA chief executive Martin Glenn said: "We understand the potential implications of our supporters' actions and wholly accept that every effort needs to be made by the FA to positively urge them to act in a responsible and respectful way."
He added: "I have not seen scenes like that in a football stadium for decades."