Scans undertaken in recent days identified the issue, which relates to the spine, but a rehabilitation programme has not yet been formulated.
Atletico Madrid said X-rays on Thursday show Costa has a herniated cervical disc.
Former Chelsea striker Costa skilled earlier this week just after showing for the ultimate 10 minutes of Atletico's three-one win about Espanyol on Sunday.
He has started eight of his side's 11 La Liga games this season, scoring two goals.
Costa has located the web two times from 15 appearances in all competitions this phrase.