The Football Association (FA) has said it is taking UEFA's warning that England, along with Russia, could be expelled from Euro 2016 in the event of further fan violence "extremely seriously."
On Sunday, the executive committee of European football's governing body warned the 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.
That has led FA chief executive Martin Glenn to urge England fans to behave and call on UEFA, European football's governing body, to improve both the segregation and the stewarding inside stadiums.
A large group of Russian fans in the stands at the Stade Velodrome in Marseille advanced on the neighbouring England section at full-time of the 1-1 draw between the sides on Saturday night, 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.
Glenn said misbehaviour by England supporters in the centre of Marseille before the game had been "embarrassing."
But he stressed that the attacks inside the stadium had been carried out by what he called "organised Russian gangs."
"We take it [the UEFA warning] extremely seriously, and we understand the implications," he told The Times.
"We are not happy about a lot of things that have gone on. We are making a plea to all England fans to remember they are in a different country and that the best way to support the team is to be patriotic but be respectful.
"The problems outside the ground were initially some unpleasant, embarrassing behaviour by English hooligans, which was stepped up massively by what appeared to be totally co-ordinated action by organised Russian gangs.
"It was shocking -- I had not seen scenes like that inside a stadium for decades.
"We were worried because we knew that England fans had been able to get tickets from the Russian website, but the segregation in the ground clearly wasn't strong enough."
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."
England captain Wayne Rooney will record a message urging fans to behave ahead of the crucial Group B game against Wales on Thursday.
Coach Roy Hodgson will also record a message before the match, which takes place in the city of Lens at the smallest of the tournament's venues.
On Sunday, UEFA opened disciplinary proceedings against Russia in connection with crowd disturbances, racist behaviour and fireworks being set off at the Stade Velodrome.
UEFA has not so far taken any action against the FA.