Scans undertaken in recent days identified the issue, which relates to the spine, but a rehabilitation programme has not yet been formulated.
Atletico said the 31-year-old requires special neurosurgery assessment before the full scale of the injury can be established.
[📋] INJURY UPDATE@diegocosta suffers a cervical disc herniation. In the next few days, our striker will be assessed by neurosurgery specialists to establish the definitive treatment.— Atlético de Madrid (@atletienglish) November 14, 2019
"The medical services of the club have conducted radiological tests of Diego Costa to assess the cervical discomfort suffered in recent days," read an Atletico statement.
"These tests have determined that the forward suffers a cervical disc herniation.
"In the next few days he will be evaluated by specialists in neurosurgery to establish the definitive treatment."
Former Chelsea striker Costa trained earlier this week after appearing for the final 10 minutes of Atletico's 3-1 win over Espanyol on Sunday.
Prior to that match, Diego Simeone's men had managed 12 goals in as many LaLiga games in a lean start to the new season.
Costa has found the net twice from 15 appearances in all competitions this term.