The French ambassador to Moscow has been summoned to the Russian foreign ministry after sharp criticism of policing at the Euro 2016 tournament, reports BBC News.
"Further stoking of anti-Russian sentiments... could significantly aggravate the atmosphere in Russian-French relations," the ministry said.
Russia's foreign minister singled out the detention of 43 fans on a bus.
France's embassy insisted that the arrest of Russia fans was "absolutely according to the law".
Meanwhile, four Russians arrested in the French city of Lille are to be deported but a match there between Russia and Slovakia on Wednesday passed off peacefully.
France's crackdown on hooliganism among supporters relates to incidents outside the stadiums.
Uefa, football's European governing body, earlier fined Russia and gave it a suspended disqualification following fan violence inside the stadium in Marseille where Russia played England on Saturday.
The Russian foreign ministry summoned Ambassador Jean-Maurice Ripert soon after a speech by foreign minister Sergei Lavrov to the lower house of the Russian parliament in Moscow.
Mr Lavrov suggested Russian fans had been provoked and criticised the way French police were subjecting them to security checks.
"It was an absolutely unacceptable incident when a bus with more than 40 Russian supporters was stopped and [the police] demanded that they leave the bus for document and ID checks," he told the State Duma.
He accused the French of violating international conventions by detaining the fans.
"It is a fact that the French behaved completely contrary to their obligations under the Vienna Convention, and I have already written to the French foreign minister, demanding that he does not allow any more such incidents to occur."
Mr Lavrov did concede that the behaviour of some Russian fans at the tournament had been poor but he added: "We cannot close our eyes to the attempts to ignore the provocative actions of other countries' fans."
In a statement on its website after Ambassador Ripert's visit to the Russian foreign ministry, the French embassy in Moscow said the French government "remained determined, together with Uefa, to stop troublemakers from ruining the party that is the Euro".
French police blamed 150 "well-trained" Russian hooligans for clashes before England played Russia.
The bus carrying 43 fans was stopped in a police operation near Cannes on Tuesday, and some of those Russia supporters are now being deported.
Among them was far-right sympathiser Alexander Shprygin, who heads a Russian football supporters' association.
Five England fans were jailed for throwing bottles at police and a sixth jailed in connection with the violence.
The match between Russia and Slovakia in Lille on Wednesday ended with the Slovaks winning 2-1.
There was no significant crowd trouble inside the stadium, although a flare was ignited by Russia fans just after their team had scored.
Lille is also being used as a base by supporters of England and Wales, who are due to play in nearby Lens on Thursday.