Koskinen is a cast member of the Finnish version of "Big Brother."
TMZ said Lambert and Koskinen were arguing inside the Helsinki tavern, then went out to the street to continue their fight.
The couple were arrested for multiple assaults after other people tried to intervene and got caught in the fray, the entertainment news Web site said.
Lambert and Koskinen spent Wednesday night in jail, TMZ said.
It was unclear if they had been released as of Thursday.
2014: The Year in Music [PHOTOS]