A brief report on the police website said the unnamed person was “one of the suspects” in the attack on January 17, which shocked the arts world.
“The necessary investigative measures are now being taken,” it added.
Sergei Filin’s eyesight was damaged when a masked attacker threw sulphuric acid in his face.
He was released from hospital in Moscow last month for follow-up treatment in Germany.
Sergei Filin will have further plastic surgery and a lengthy convalescence.
The flat of a Bolshoi ballet dancer, soloist Pavel Dmitrichenko, was searched by Moscow police on Tuesday, a separate brief report on the police website said. There was no suggestion of a link to the detained suspect.
The attack on Sergei Filin came after a period of infighting at the Bolshoi, one of Russia’s most prestigious ballet companies.
Unnamed sources quoted by Russian media say the suspect is a man who was detained early on Tuesday in a town near Moscow. He is not connected to the Bolshoi, the Russian website Life News reports.
Searches are reported to be under way at the suspect’s home and other addresses connected to him.
Sergei Filin, 42, was badly burned during the attack, which happened in the street as he returned to his home in Moscow late at night.
It appears the assailant called out to the artistic director by name, then splashed him before fleeing from the scene.
A security camera apparently captured grainy footage of the event.
The artistic director said later he was certain who was behind the attack, but would only speak after an announcement by investigators.
Sergei Filin argued the aim of the attack had been to remove him from his post, adding that he had long been the target of threats.
Police have questioned several Bolshoi employees as potential witnesses.
They are treating the attack as a premeditated act of grievous bodily harm, motivated by Sergei Filin’s professional activities.
German doctors think they can restore some of Sergei Filin’s sight.
A spokeswoman for the Bolshoi said they hoped Sergei Filin would return to the company for their trip to London this summer.
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