The Bayern Munich midfielder had surgery earlier this month after he aggravated an existing issue while on Germany duty.
Bayern initially gave no timescale for Goretzka's recovery but Kovac does not expect his return to be imminent, though there is better news on left-back David Alaba.
"David Alaba has trained today, it looks good," Kovac told reporters ahead of Saturday's Bundesliga trip to bottom side Paderborn.
"Let's see how it develops tomorrow [Friday] with him. Leon Goretzka still takes a bit longer.
"I expect [he will return] only after the international break."
Kovac had previously stated on Alaba, when suggesting that he may be forced to sit out a meeting with Spurs: "Obviously, David is a really important player for us. Unfortunately he's going to be out for a couple of weeks but Lucas Hernandez won the World Cup in that position.
"He has a different interpretation of how to play at left-back than David does but he does a very good job.
"Alphonso [Davies] can play in that position, that's actually why we fielded him against Leipzig. He didn't play as a left-back but we fielded him so that he could work together with Lucas.”
Meanwhile, Jann-Fiete Arp is yet to make his Bundesliga debut for Bayern after joining from Hamburg and the striker's wait will continue after he broke a bone in his hand during training.
"We will have to wait on what the doctors say before we can determine when he will return to full training again," Kovac added of the 19-year-old.
"It is difficult. You have to be careful that the bone grows together again. Shocks and jolts could complicate things, so we need to see what the doctors say so we know when to train again.
"Fiete has gotten the time from us and will continue to get it. We try to make him better every day in training, but you cannot expect too much from him.
"He may have been hyped up a bit too much at a young age, you have to be patient with him."