The striker was withdrawn 30 minutes into Tuesday's 1-1 draw at Barcelona in the first leg of Atleti's UEFA Champions League quarter-final.
Costa had been a doubt for the encounter with a knee problem, but it was a hamstring issue that brought his evening to a premature end, although he has avoided a tear.
The club are yet to put a timescale on the the 25-year-old's recovery, but there is considerable doubt over his involvement in the second leg in Madrid next Wednesday.
The Spain forward has played a huge part in Atleti's rise to the top of La Liga this season, scoring 25 goals in 30 appearances.
Diego Simeone's men host Villarreal in the league on Saturday.
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