The forward has been out since January and went under the knife on Wednesday after a scan revealed the injury had not healed as well as hoped.
Kozak's spell on the sidelines is likely to be extended by a number of weeks after boss Paul Lambert confirmed the setback.
"He had an operation on Wednesday," he said
"They had to go back into his leg and put a little pin in it
I haven't spoke to the doctor about the length he will be out but it's going to be a while.
"It's a setback for him
The doctors are with him so we'll find out soon (how long he's out for).
"He wasn't too far away from being back
He was running and doing everything
It's a blow for him and blow for us.
"It's just not healed the way we thought."
Kozak cost Villa £7million when he signed from Lazio last summer and had scored four goals in 15 games before suffering the injury in training after a collision with defender Ciaran Clark.
He was expected to return soon and Lambert admitted the news had affected the 25-year-old Czech Republic international.
"It's a nightmare
I spoke to him about his injury and he felt a bit down which is understandable because he's worked so hard to get back," he said.
"When he came his goal ratio was very good
He's a handful
He's just been unfortunate that he's suffered that setback."
Captain Ron Vlaar is also out of Saturday's Barclays Premier League visit of Manchester City with a calf injury, something Lambert attributes to World Cup burn-out after Holland's run to the semi finals.
"Hopefully he will be back after the international break, it's just a calf injury," he said
"He's had it for the last couple of years
He did his right, then his left
I knew it would happen, just never this early.
"He looks after himself in the best way he can
It was just fatigue from the World Cup."
Source : PA