The Northern Ireland international has been out of action since he picked up the problem during his side's 1-1 draw with local rivals Aston Villa on September 30.
"I'm not too sure what the overall timescale on the injury is, but it's coming along," Brunt told the Birmingham Mail.
"It wasn't as bad as we thought at first because it felt like a nasty one when I got it at Villa Park, so in that sense it isn't too serious.
"The problem is that it's happened a couple of times now, so this time I want to make absolutely sure it's right before I come back because it's no use to anybody doing two or three weeks at a time and then getting another injury."
The Baggies return to Premier League action following the international break on Saturday when they host champions Manchester City at the Hawthorns.