The latest violence followed two nights of trouble in the French port city.
French police used tear gas for a third day and a water cannon. They say one Briton was seriously hurt in clashes.
There were also clashes in the stadium following England's 1-1 draw with Russia, after Russia fans appeared to rush at the England supporters.
Witnesses said flares had been let off by Russian fans during the game, and some fans had then climbed across barriers designed to keep rival fans apart.
BBC Sports editor Dan Roan said England fans began jumping over the perimeter fence in panic.
"Big questions for UEFA here," he tweeted. "Stampede appears to have been triggered by explosion. How did such a device make it inside and where was segregation?"
The FA condemned the violence saying they were "very disappointed by the terrible scenes of disorder".
A police spokesman said the Briton who was injured in the clashes before the match had been resuscitated by officers, but gave no further details.
A Foreign Office spokeswoman said: "We are in contact with French authorities about a British national injured in Marseille and stand ready to provide further support."
Before the game fans had walked past ranks of police officers in riot gear on their way to the Stade Velodrome venue.
The great majority of fans appeared to be peaceful but there were reports of a few scuffles outside the stadium.
European football's governing body Uefa said it "firmly condemns the incidents in Marseille", adding that "people engaging in such violent acts have no place in football".
Earlier on Saturday, shadow home secretary Andy Burnham had said England fans' behaviour was "embarrassing", given the terrorism threat to France.
BBC News correspondent Nick Eardley, who is in Marseille, described the situation as "pretty terrifying" as fans fought with Russia supporters.
He said he had witnessed bloodied England fans being "dragged away by police", to the sound of sirens blaring out across the city's main square.
Police told the BBC that six people had been arrested and at least five had been injured during the scuffles.
Local police chief Laurent Nunez told AFP news agency: "Police intervened in a clash between English, Russian and French supporters in the Vieux Port district."
Some British journalists in the area have described the scenes on social media, and pointed the finger at a gang of black-clad Russians, who they said had entered the port city's main square from a side street apparently seeking violence.
From the scene
Image copyrightGETTY IMAGES
By Nick Eardley, BBC News correspondent
England and Russian fans have been gathering at the city's port all day; I was mingling with them earlier and many were in good spirits - the odd tense scene, but others taking pictures and joking with police.
It's unclear what actually caused things to turn; one police officer told me there had been a confrontation between England fans and Russian fans in the square.
Some outlets have reported French police saying there were French fans involved too.
What followed was a pretty terrifying exchange of bottles being thrown shortly after 16:00 local time (15:00 BST).
I was in the square at the time and there were fans running everywhere after the first tear gas was fired, many clearly getting injured in the clashes. That went on for some time.
A group of fans - and from where I was standing, just behind the police line, the vast majority appeared to be England fans - continued to throw bottles at police, with officers returning volleys of tear gas and eventually charging those fans left in the square.
It's hard to say exactly what happened after that, but I saw fights in a number of streets around the port. It was only around an hour ago when I left that part of town that things seemed to be getting back to normal, though obviously with a considerably bigger police presence.
Meanwhile, Gary Lineker tweeted: "The trouble in Marseilles is utterly sickening."
Earlier, Mr Burnham said England fans had been "let down by a minority", who may have been provoked but were "not blameless" over the first two nights' violence.
In a series of tweets, Labour's Mr Burnham said when the terror threat in France was taken into account, it made the "behaviour of these England 'fans' even more embarrassing".
Dave Tomlinson, from the Football Supporters' Federation, who is in Marseille, told the BBC it had been "disappointing to see any problems" involving England supporters.
"We understand from the reports that it's a very small minority of England fans that have been involved," he said.
He also said the style of policing was "a little bit different" from what England fans would be used to.
"The first option at the moment seems to be that the French police are going to put tear gas in the areas where there are potentially going to be problems."