UEFA's executive committee told the football associations of both countries Sunday that their teams could face further sanctions -- including potential disqualification from the tournament -- if their fans were involved in a repeat of the scenes that marred their fixture in Marseille Saturday.
UEFA, the governing body of football in Europe, has already opened disciplinary proceedings against the Russian Football Union over its supporters' behavior in Marseille.
It faces charges of crowd disturbances, racist behavior and setting off fireworks Saturday in the southern French city's Stade Velodrome.
"UEFA expresses its utter disgust for the violent clashes that occurred in the city center of Marseille, and its serious concern for the incidents at the end of the match inside Stade Velodrome," UEFA said in a statement.
"This kind of behavior is totally unacceptable and has no place in football."
It said a decision on the sanctions would be made "within the next few days," once evidence had been considered.
No action has been taken against England's Football Association, whose fans were also involved in the clashes on the third straight day of football-related violence in the port city. UEFA is responsible for incidents at match venues, but not in surrounding areas.
Dozens injured, 8 arrests
At least 35 people were injured Saturday, four of them seriously, said Mathieu Duroselle, a spokesman for Marseille police. The number included three officers who suffered slight injuries, he said.
Eight people were arrested, including British, Russian and French citizens.
The British government said in a statement Sunday that six British nationals were hospitalized and several others had received treatment.
"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," it said. "We condemn the violence and urge calm on all sides."
It said it welcomed UEFA's launching of an investigation into the violence at the stadium, and had offered to send additional British police to France ahead of England's next match, against Wales in Lens.
FIFA, the sport's international governing body, also released a statement condemning the violence.
The organization said it was "wholly unacceptable to see such shameful scenes around football matches perpetrated by a minority of idiotic troublemakers, who have nothing to do with football and its true fans."
The specter of hooliganism returned to the high-profile tournament Saturday as rival fans brawled before and after the match.
The fixture, anticipated by fans as the marquee game of the second day of Euro 2016, ended in a 1-1 tie.
In chaotic scenes involving hundreds of fans, throngs of rival supporters rushed at one another, hurling bottles, chairs and other objects, and forcing police in riot gear to fire tear gas
Distressing footage emerged of men kicking and stomping on another person lying on the street in broad daylight, as the rivalry turned violent prior to kick-off.
Then, at the end of the game, Russian fans charged at the section of the stadium containing England supporters and more chaos ensued. Smoke filled the streets as panicked fans fled.
Rebekah Vardy, wife of English striker Jamie Vardy, was critical of the crowd policing.
"That has to be up there with the worst experience EVER at an away game! Teargassed for no reason, caged and treated like animals! Shocking!" she wrote on her Twitter account, calling the scenes "horrific and uncalled for."
In a statement, UEFA acknowledged "segregation issues" at the stadium and promised to "implement corrective measures to strengthen the deployment of security personnel at stadiums, in close collaboration with local authorities."
France deploys extra security
Both teams have reputations for having hooligan elements among their support base, and officials had identified Saturday's match as a potential flashpoint.
At the last UEFA European Championship, played in Poland in 2012, Russia was penalized for the poor behavior of traveling supporters -- including violent disorder and the racial abuse of a black player from Czech Republic
France's Interior Ministry classified Saturday's match as "at risk" and police sent in reinforcements, with about 1,000 police officers deployed to provide security, according to Marseille police spokesman Duroselle.
The French Interior Ministry had previously announced international moves to keep hooligans out of the country during the tournament, which is also being held under the shadow of a potential terrorist threat.
At the request of French authorities, Britain retained the passports of thousands of people considered hooligans, stopping them from coming to France during the competition.
"There have been 3,000 entry denials by the Minister of Interior, based on lists of people who are banned from stadiums in different countries participating in the competition, or individuals likely to disrupt the order in national territory," the Interior Ministry said.
Two hundred foreign police officers had also traveled to France to help prevent any outbreaks of violence during the tournament.
Bars closed early
Other measures to prevent violence included bars closing early in the Vieux-Port area in Marseille -- the same area clashes broke out Thursday and Friday nights among football fans.
Multiple bars in the area said they shut their doors at 8 p.m. local time Saturday.
Among the bars is O'Malley's, an Irish pub that usually closes at 11 p.m.
CNN law enforcement analyst Steve Moore said that violence among fans had become less of a feature of football tournaments in recent years, as the game's administrators looked to stamp it out of the sport.
The scenes in Marseille, he said, "were broadly in line with what we would have expected about 15 years ago."
Skirmishes were also reported Saturday night in the French city of Nice ahead of Sunday's match between Poland and Northern Ireland.
The situation in Paris for Sunday's game between Croatia and Turkey was largely calm, according to Paris police spokeswoman Johanna Primevert. She said there were isolated incidents of local troublemakers throwing projectiles near the Parc Des Princes stadium in an attempt to instigate visiting fans, but police swiftly brought the situation under control.