Sean McBride, aged 66, serving a short sentence on tax and insurance evasion charges, died over a week ago at the Curragh prison.
It was first thought Mr McBride died from a heart condition.
But a garda investigation was launched after another inmate claimed that the elderly man had been poisoned.
Six detectives from Newbridge, in Co Kildare, went to the prison last Monday to interview a number of inmates, including some convicted of sex offences.
The State pathologist was called in to examine the body and toxicology tests will be carried out to determine whether he was poisoned.
A murder investigation may then be launched.
Results from the toxicology tests are expected to take up to two weeks.
Garda headquarters yesterday confirmed an investigation, headed by Superintendent John Murphy from Newbridge, has begun into the circumstances of the death. A spokeswoman said the man died on Thursday, February 13.
After further information was received on February 16, the State pathologist was called in.
A prison source said gardaí came to the prison after the man died, but this was standard practice.
“They came back in force last Monday after the allegation or suggestion was made that bleach was put in his mineral.
“They interviewed some of the inmates.
“The information came from a prisoner. If it comes back that there was a bleach in his system, it’s going to be a murder investigation.”
The Prison Service said the inmate had a history of heart problems and that was thought to have been the cause of his death.
“It was genuinely felt it was related to his medical condition, but since then, there was further information his drink may have been tampered with.”
The vast majority of Curragh inmates are sex offenders and about 10% are priests and brothers.
A small number of prisoners are convicted of various other offences.