While not identifying Spacey directly, it is understood that police are looking into allegations that he sexually assault a man in Lambeth in 2005. It said numerous alleged incidents took place at the theatre - located in the borough of Lambeth - and involved members of staff.
But it said the same man was accused November 1 of another assault in 2008, also in Lambeth.
The theatre launched an investigation into its former leader last month after American actor Anthony Rapp alleged that Spacey had climbed on top of him on a bed when Rapp was 14 and Spacey 26.
The Old Vic's announcement follows recent allegations of sexual harassment and predatory behaviour made against the double Oscar victor and former House of Cards actor while at the theatre and elsewhere in the entertainment industry.
London police confirmed toThe Hollywood Reporter that the complaint was filed Friday, just a day after the Old Vic theater revealed it had received 20 accusations of sexual misconduct regarding the Usual Suspects actor.
The Met Police is already investigating one claim of sexual assault by a different male victim, which occurred in Lambeth, where the theatre is based, in 2008.
In response to the claims, Spacey issued a post on Twitter in which he wrote that he did not remember the alleged incident and also announced he is living as a gay man. The Old Vic said a "cult of personality" around Spacey had discouraged people from speaking out.
There were many other allegations made after Rapp's story became public.
The alleged victim is believed to be an aspiring actor who sought career advice from the House of Cards star.