Atletico can bring in Kondogbia outside the transfer window because La Liga rules grant a club 30 days to find a replacement when a player's buyout clause has been paid.
The midfielder first shone in La Liga at Sevilla in the 2012-13 season, before moving to Monaco and Inter, and joining Valencia in 2017.
His departure is acrimonious, after Kondogbia accused Valencia and president Anil Murthy of reneging on an agreement to allow him to follow other key players such as Dani Parejo, Rodrigo Moreno and Ferran Torres in leaving Mestalla this summer.
"Having destroyed an ambitious project, you had to mislead your coach and finally me. Thank you, Anil Murthy," he posted on Instagram last month.
Valencia announced the player's departure with a curt, one-line statement, tweeting: "So long, Geoffrey Kondogbia."