The 33-year-old developed the problem after taking the winning penalty against Port Vale in Tuesday night's pre-season friendly.
Rowett told RamsTV: "[Bent] went to see a specialist in London after the injury at Port Vale and it's not good news.
"He's probably going to be out up until around Christmas time so it's a little bit of a blow. He's looked sharp in the games so far but it's something we're going to have to deal with."
Bent, a former England international, made 42 appearances for Derby last season, scoring 13 goals.