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Costa was taken off early in the second half against Sporting Lisbon on Thursday as Atletico booked their place in the last four of the competition with a 2-1 aggregate victory.
The LaLiga club were drawn against Arsenal in the semis before announcing later on Friday that Costa had suffered “a hamstring injury to his left leg”.
Spain international Costa underwent tests in Madrid but the length of the lay-off was not determined in a statement published on their official website.
The Gunners, currently sixth in the Premier League, host Diego Simeone’s side on April 26 before the second leg at the Wanda Metropolitano Stadium on May 3.
Atletico are second in LaLiga, 11 points behind leaders Barcelona with seven games remaining, and face Levante at home on Sunday.