The former England forward suffered the injury in last week's pre-season fixture against Port Vale.
Bent has scored 28 times in 84 Derby games and news came after Saturday's friendly win over Kaiserslautern.
"Gutted to be having a spell on the sidelines. Great result for the boys yesterday. Long may it continue," the 33-year-old said on Twitter.
Bent's injury was confirmed by a scan and boss Gary Rowett told the club website it was "a bit of a blow".
He added: "He had looked sharp in the games so far but it's something we will just have to deal with."
Derby County's Darren Bent is likely to be out until around Christmas because of a hamstring injury.