The victim was treated for a facial injury following the incident on Tuesday.
Staffordshire Police said officers were called to a "report of an assault between two prisoners".
The category C prison and young offenders institute has a population of around 600.
A force spokeswoman said: "A man was injured and taken to hospital where he was later released a short time later.
"Following further discomfort, the man was taken to Queen Elizabeth Hospital in Birmingham this morning with a facial injury and discharged again.
"HM Prison Service are currently looking into the matter."