Villa limped out of Sunday's 4-0 La Liga victory over Real Sociedad soon after opening the scoring in the 38th minute and underwent tests on the problem on Monday.
Those tests have revealed the extent of the injury with Atletico tweeting: "David Villa suffered an extension/strain without tear of the biceps femoris muscle at the back of the left thigh."
Atletico have not said how long the 32-year-old is likely to be out for, but reports in Spain suggest he could be sidelined for around 10 days, which would rule him out of Wednesday's semi-final first-leg against Real at the Bernabeu and possibly next Tuesday's return leg at the Vicente Calderon.
Atletico are the Copa del Rey holders having beaten Real in last year's final - ending a 14-year winless run against their bitter rivals.
Villa, Spain's record goalscorer, has scored in Atletico's last three league games to help them open up a three-point lead at the top of the Liga standings over reigning champions Barcelona.