Simeone, 50, is currently isolated at home and has not shown any symptoms.
Atletico's squad and staff returned early from a pre-season trip just outside Madrid after a test on Thursday came back positive. Further tests on Friday showed Simeone had caught the coronavirus.
"The analysis of these new samples in the laboratory has determined that our trainer, Diego Pablo Simeone, has given a positive result for Covid-19," read the club statement.
"Fortunately, our coach does not present any symptoms and is at his home isolated and fulfilling the corresponding quarantine after being with the team since last Monday."
Simeone was absent from training on Saturday. Atletico's first La Liga match against Granada is not scheduled to be played until September 27 as the club have been given extra time off before starting the season.
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Heartbreak for Atletico Madrid in Champions League
Atletico Madrid's Saul Niguez reacts following his team's Champions League quarterfinal loss to RB Leipzig at the Jose Alvalade Stadium in Lisbon. AP
Atletico Madrid's Joao Felix reacts after the second goal by RB Leipzig during the Champions League quarterfinal at the Jose Alvalade stadium. AP
Atletico Madrid's goalkeeper Jan Oblak fails to stop a shot by Leipzig's Tyler Adams during the Champions League quarterfinal on Thursday. AP
Atletico Madrid's Joao Felix scores from the penalty spot during the Champions League quarterfinal against RB Leipzig. AP
Leipzig's Yussuf Poulsen consoles Atletico Madrid's Jose Gimenez, right. AP
Tyler Adams, left, of Leipzig celebrates with teammate Konrad Laimer. EPA
Goalkeeper Jan Oblak of Atletico after Leipzig scored their second goal. EPA
Leipzig's Yussuf Poulsen, right, and teammate Kevin Kampl celebrate after their team's Champions League quarterfinal win over Atletico Madrid. AP
Atletico Madrid's head coach Diego Simone gestures in the final minutes of the Champions League quarterfinal against RB Leipzig. AP
RB Leipzig's coach Julian Nagelsmann and Yussuf Poulsen celebrate after the match. Reuters