It follows findings in the Roma report, written by the Ombudsman for Children, Emily Logan.
A chief superintendent has been appointed to lead the inquiry, which will try to examine communications between gardaí and journalists in relation to Child T.
The seven-year-old girl was removed from her family in Tallaght on October 21 last.
A Garda spokesman told the Irish Examiner: “Immediately following the publication of the report by the Ombudsman for Children, the commissioner appointed a chief superintendent to investigate issues raised in the report relating to personal and sensitive information appearing in the public domain.”
Ms Logan said media coverage generated “entirely unwarranted and distressing notoriety for Child T’s family”. She said an article published on October 22 contained “striking similarities” with an internal Garda report on October 21, which “leads the inquiry to believe on the balance of probabilities that the information in question came from someone within An Garda Síochána”.
“The inquiry therefore believes that the release of details regarding Child T into the public domain may have been the result of a breach of discipline and/or an offence.”
It is understood chief superintendent Francis Clerkin of Dublin northern region is conducting the investigation. His team will try and determine which gardaí spoke to what journalists on October 21.
It is thought the investigation is under disciplinary regulations, but sources said it could be upgraded to a criminal investigation, depending on what emerges.