The French government has urged cities hosting Euro 2016 matches to ban alcohol near venues and fan zones.
It follows three days of clashes between fans and police in Marseille.
Uefa is investigating after Russia supporters appeared to rush at England fans in the stadium after their draw on Saturday. It has threatened to ban both teams if there is further fan violence.
The UK government has offered to send extra British police ahead of England’s next match in Lens on Thursday.
French Interior Minister Bernard Cazeneuve said: “I have asked for all necessary measures to be taken to prohibit the sale, consumption and transport of alcoholic drinks in sensitive areas on match days and the day before, and on days when fan zones are open.”
The ban will include public areas, as well as shops and off-licences, Mr Cazeneuve said.
Senior local officials can also ban bars and cafes from serving drinks on their terraces in containers that can be used as missiles.
“The events which took place in Marseille… are unacceptable. Unacceptable for the authorities, unacceptable for society, unacceptable for football lovers,” Mr Cazeneuve said.
Some Russia supporters had smuggled in flares which they set off at the end of the match
A number of Russia supporters appeared to kick and punch fleeing England fans, who were forced to clamber over fencing to escape.
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, the country was under intense scrutiny.
Russian and English supporters as well as French locals also clashed in pitched battles for three days in Marseille ahead of their teams’ first-round encounter.
As many as 20 England fans were injured, with a number taken to hospital, and at least one Briton is understood to be in a critical condition.
Uefa said it had not taken any action against the English FA, because skirmishes involving England fans before kick-off fell outside its remit.
A number of Russia fans rushed towards England fans at the end of the match
FA chief executive Martin Glenn said a letter from Uefa about fans’ behaviour, was being treated with the “utmost seriousness”.
“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,” he said in a statement.
“Violent scenes like those witnessed over the weekend in Marseille have no place in football, nor society as a whole.”