Costa will miss Atletico's Champions League game at home to Lokomotiv Moscow on Wednesday and is expected to be out for at least a month.
That would mean the 32-year-old also sitting out La Liga games against Valencia and Real Madrid in the next few weeks, as well as the last two Champions League group fixtures against Bayern Munich and RB Salzburg.
Atletico sit second in Group A after taking four points from their opening three games.
"Diego Costa underwent tests this morning that confirmed he has suffered a spontaneous deep vein thrombosis in the right leg. It is not related to trauma or previous injury," a statement from Atletico read.
Costa has just recovered from a separate muscle injury that has caused him to miss Atletico's last four matches.
Diego Simeone is already short of forwards, with Luis Suarez sidelined due to a positive test for coronavirus.