The 27-year-old joined the Villans from Lazio in a £7m deal in September 2013 but suffered a broken leg four months later and has struggled for fitness ever since.
The Czech Republic international has made just 22 appearances for the Midlands side and is now likely to leave the club when his contract expires this summer.
"He has had an ankle operation and he's out for the rest of the season," Villa boss Steve Bruce told reporters today. "To be honest with you he's had a really bad time.
"It's a shame because he's been hit by so many injuries. He's a good player but he's out of contract in the summer."
Kozak's last appearance for the club came as a substitute in a 3-1 loss at Bristol City on August 27.