Uefa has threatened to disqualify England and Russia from Euro 2016 if there is any further violence by fans.
It has begun disciplinary proceedings against Russia – but not England – after “totally unacceptable” scenes at Saturday’s England-Russia game.
Footage appeared to show Russia fans rush at England supporters after the Euro 2016 1-1 draw in Marseille.
Charges against Russia are for crowd disturbances, racist behaviour, and setting off fireworks, Uefa confirmed.
There were issues with segregation of fans, the body admitted, promising security would be “strengthened”.
The UK government said it had offered to send extra British police to France ahead of England’s next match in Lens on Thursday.
Witnesses said trouble began in the stadium after flares were let off by Russian fans near the end of the game. Some then climbed across barriers designed to keep rival fans apart.
A number of Russia supporters appeared to kick and punch fleeing England fans, who were forced to clamber over fencing to escape.
Earlier in the day, England, Russia and local fans were embroiled in ugly clashes with riot police, with a number of bloodied fans being taken to hospital.
In a strongly-worded statement, European football’s governing body said it would not hesitate to impose sanctions on either Russia or England should such violence occur again.
Measures included the potential disqualification of their respective teams from the tournament, it said.
Earlier Russia’s sports minister Vitaly Mutko said Uefa had “done the right thing” to start a disciplinary case against the Russian Football Union, R-Sport news agency reported.
“There were flares, there was a flare gun, there had been clashes in the stands, it’s necessary to sort all of this out,” he reportedly said.
“The fans were rooting greatly, but there are people who do not come for the football.”
Also on Sunday, French Interior Minister Bernard Cazeneuve said alcohol restrictions near “sensitive” venues would be extended, with sales bans on match days and the eve of match days.
England spectators who were at the match have described a “desperate scramble” to escape charging Russia fans, some of who were wearing gum shields and MMA (mixed martial arts) gloves, one witness claimed.
Cory McLeod, who was in the stadium, told the BBC: “A wall of Russian fans charged at our section, which had kids, women, and even some Russian fans in.
“I couldn’t see if any of them had weapons or anything but people were running fast, it was manic, just a desperate scramble towards a large fence in the corner.”
Others spoke of “slack” security. Sam Blackwell, from Kingston upon Thames, south-west London, said the only entry search was “a quick pat down”.
“This would go a long way to explaining how the Russian fans got flares, fireworks and whatever the big bang was into the ground.”
Six British nationals remain in hospital and several more have received treatment, the UK government said.
Martin Glenn, FA chief executive, described the violence as “shocking”, adding: “I’ve not seen scenes like that in a football stadium for decades.”
He said: “We were worried because we knew for a while that England fans had been able to get tickets from the Russian website, but the segregation in the ground clearly wasn’t strong enough to stop the terrible scenes that we witnessed.
“So it was very upsetting, it must have been really, really, scary for the people involved.”
Sanctions against Russia will be decided at a disciplinary meeting on Tuesday, Uefa said.
BBC sports editor Dan Roan said as hosts of the 2018 World Cup, Russia was under intense scrutiny.
Uefa’s decision would also come under scrutiny, he said, with Russia’s next match against Slovakia taking place in Lille on Wednesday – the day before England’s match against Wales just 24 miles away in Lens.