The HSE is being asked to establish the full extent of a practice of double burials.
It has emerged that a number of babies bodies were buried with deceased adults, although the practice is thought to have ended in the 1980s.
Sinn Fein's Health spokesman Caoimhghín Ó Caoláin says the HSE must establish the full extent of the practice and report to the Minister for Health.
“While I accept that those responsible meant no harm by their actions, their actions were nonetheless inappropriate and likely offensive to both the family of the dead infant and the family of the deceased adult.
“I believe it is important that the HSE now establish the full extent of this practice and report to the Minister for Health who should then advise if further steps are necessary,” Mr Ó Caoláin.
While Fianna Fail's Billy Kelleher says it must be determined what hospitals were carrying out this practice, how long it went on for, and if family members of both deceased notified of the burial procedure.
Medical journalist Priscilla Lynch says there may be a number of reasons for this type of burial, including social, religious and financial.
One possible reason for the practice could have been the desire to bury a child that was not baptised in concentrated ground.
“It would have been a social and religious issue - possibly the babies would have been stillborn or would have been miscarried,” Ms Lynch said.
“They wouldn’t have been baptised so burying them with an adult would have been a way to ensure that they were buried in concentrated ground,” she added.
Another factor could have been financial, with unmarried young mothers potentially unable to bear the cost of a separate burial.
“Perhaps the mother would have been unmarried,” Ms Lynch said. “And poverty as well in the case of the cost of the funeral as well, it would have been seen as a way of burying the infant without incurring the cost,” she added.