Taylor sustained the injury during the friendly clash with Nottingham Forest at the City Ground in early August.
The 29-year-old has already missed City's opening four Premier League games but is hopeful of being back in contention in three weeks.
Taylor said: "The injury is going quite well. Obviously it's frustrating to miss the start of the season but I think it's been about five weeks now and we've probably got about three weeks to go.
"It's a really frustrating time for me. I'd had a good pre-season and I felt really fit and raring to go for the start of the season.
"Then not even two weeks before the start of the season I pick up an injury that keeps me out for a couple of months.
"I'm going to have to do another pre-season-type programme now to get back."
Meanwhile Cameron Jerome has confirmed he is making slow progress after picking up an ankle problem against Portsmouth at St Andrews three weeks ago.
He said: "The injury is coming on all right at the moment. It's just steady rehab. I'm a couple of weeks away yet before I can step it up with running and get back into training.
"It's a matter of taking my time, strengthening it up and letting it heal.
"I've missed a couple of games already and, with this international break, that gives me an extra week for me to get fit and stronger.
"Hopefully I can come back as quickly as possible and it won't be so long in rehab."