One of the men had an argument with a police officer, after which he was detained, according to a source.
The other man hurt his hand and has since been discharged from the hospital.
British Embassy in Moscow confirmed that British citizen was detained in Russia after incident in the train to Volgograd.
According to a post on Russian social media, purporting to be a leaked internal railway operator's report, said that the injured fan had broken a window on board the train and received cuts to his hand.
"Cases of administrative offenses were initiated against both foreigners for appearing in public places under the influence of alcohol. Besides, a case of administrative offense was initiated against the citizen who refused to follow the lawful demands of a police officer," police said.
It was not immediately clear when the incidents on the train happened.
The first FIFA World Cup in the Russian history will continue until July 15. The cities of Moscow, St. Petersburg, Rostov-on-Don, Sochi, Yekaterinburg, Saransk, Kaliningrad, Nizhny Novgorod, Kazan, Volgograd, and Samara host the matches of the contest.