England and Russia could be disqualified from Euro 2016 if their supporters are involved in further violence, UEFA has said.
UEFA had already opened disciplinary proceedings against Russia over the behaviour of their supporters at the Stade Velodrome in Marseille on Saturday night.
There were ugly scenes in the city before and after Saturday's match involving English and Russian supporters and locals, and six British nationals remain in hospital.
On Sunday afternoon, UEFA warned both nations that if their supporters are involved in more violence, the teams could be thrown out of the competition.
A UEFA statement said: "The UEFA Executive Committee would like to express its disgust for the violent clashes which occurred in the city of Marseille.
"Such unacceptable behaviour by so-called supporters of the national teams of England and Russia has no place in football, a sport we must protect and defend.
"The UEFA Executive Committee has warned both football associations that - irrespective of any decisions taken by the independent disciplinary bodies relating to incidents inside the stadium - 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.
"We urge both the FA and the RFU 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."
There are fears Russian and England supporters could clash again in Lille later this week. Russia play Slovakia in Lille on Wednesday, and England play in Lens - 20 miles away - on Thursday. But many England fans are staying in Lille.
UEFA to discipline Russia
UEFA has opened disciplinary proceedings against Russia over the behaviour of their supporters
France's interior minister Bernard Cazeneuve has proposed that host cities ban the sale of alcohol the day before and the day of matches, in "sensitive" areas such as fan zones.
"It is absolutely necessary that the national federations whose supporters create incidents of this nature be punished for what happened inside the stadium, and also outside," Cazeneuve said.
The Football Association's chief executive Martin Glenn, speaking before UEFA's threat of a ban, 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.
"Violent scenes like those witnessed over the weekend in Marseille have no place in football, nor society as a whole.
"We want people - fans and locals - to feel safe and enjoy a festival atmosphere at the Euros and we will continue to work closely with all the relevant authorities to achieve that."
"Russia have already been charged for the events in the ground so we will let UEFA decide on that and they are clearly holding the Russians to account on that one.
"Clearly there has been a problem [with England fans]. There is anti-social behaviour by a minority of supporters and on top of them we have seen a level of criminality from organised Russian gangs and locals, which has changed the scene totally."
A UK government spokesman said: "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. We condemn the violence and urge calm on all sides.
"These incidents go directly against the spirit of the tournament and are the actions of a minority. The vast majority of fans in France simply want to enjoy the football.
"So we welcome UEFA's decision to launch an investigation into the violence inside the stadium in Marseille last night, and stand ready to assist in any way we can. We appreciate the Russian Sports Minister's statement of support for the investigation."