The Prison Officers Association (POA) said there was an "ongoing disturbance" at the category B training prison, HMP Swaleside, on the Isle of Sheppey.
Specialist "Tornado" teams have been deployed to tackle the unrest, the Prison Service said.
POA chairman Mike Rolfe said the extent of the disturbance was unclear, but it did not appear to be on the scale of a riot at HMP Birmingham last week.
Riot police were used to quell last Friday's disturbance, which lasted for 12 hours and involved hundreds of inmates.
A Prison Service spokesman said: "The Prison Service is dealing with an incident involving 60 prisoners at HMP Swaleside.
"The incident is contained to one landing on A Wing. The rest of the prison is secure with all prisoners in locked in cells."
Later, he added: "The situation at the prison remains contained and is in the process of being resolved."
Mr Rolfe described Swaleside as "a particularly difficult place to work" because of an "acute staff shortfall" and high levels of violence.
A man claiming to be an inmate there contacted the BBC to say the disturbance had started when staff raided cells and confiscated prisoners' possessions.
The unverified source said tension between inmates and prison officers had increased after an intake of young, inexperienced staff, and the cutting of family visits.
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