Meanwhile, French authorities said they had detained 63 people linked to violence since the start of Euro 2016 and gave police new powers to ban alcohol around stadiums.
European football's governing body UEFA issued the warning as German hooligans became the latest to join street battles that have marred the tournament, following three days of clashes in Marseille between Russian and English fans.
Tensions between supporters of the two countries came to a head inside the Stade Velodrome after Saturday's 1-1 England-Russia draw.
With more high-risk matches looming, the UEFA executive committee expressed "disgust" at the clashes.
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UEFA has laid charges against Russia, accusing their fans of crowd disturbances, racist behaviour and throwing fireworks and missiles in the Marseille stadium.
A decision on sanctions is to be announced Tuesday.
The UEFA executive committee also said "it will not hesitate to impose additional sanctions" on the countries, "including the potential disqualification of their respective teams from the tournament, should such violence occur again".
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But the European body did say that security has to be improved.
"There were segregation issues at Stade Velodrome and (UEFA) will implement corrective measures to strengthen the deployment of security personnel at stadiums."
The French government gave police chiefs powers to ban alcohol sales and consumption from the day before matches near the 10 stadiums and fan zones being used for the month-long, 24-nation tournament.
The interior ministry said Sunday that 116 people had been questioned in relation to violence since the start of the tournament, of which 63 were taken into custody.
There were new violent incidents Sunday, as German football fans clashed with Ukrainian supporters ahead of their Group C game in Lille.