Further tests are being carried out, but if confirmed, it would be the first case of BSE (Bovine spongiform encephalopathy) in Ireland since 2013.
The animal involved is a five-year-old cow from a dairy farm in County Louth.
The Irish Department of Agriculture, Food and the Marine (DAFM) said the cow was not presented for slaughter and did not enter the food chain.
In a statement, a DAFM spokesperson said: "The case was identified through the department's on-going surveillance system on fallen animals (that is, animals which die on farm)."
They added that the results of the further tests should be available next week.
The Irish Minister for Agriculture, Simon Coveney, said this appeared to be an isolated case, and there was absolutely no risk to people.
A positive result would be a blow for the Irish beef trade, which just last week welcomed the decision by the World Animal Health Organisation to grant the Republic of Ireland a "negligible risk" status in respect of BSE.
At the time, Mr Coveney hailed it as a "landmark decision" and "major step forward on BSE certification".
He said the new status had "reflected the huge progress made over many years in eradicating this disease from the national herd".
Stormont's Agriculture Minister Michelle O'Neill said her department had been in contact with their southern counterparts and was being kept informed of the situation.
"We are now awaiting details of the confirmatory tests following DAFM's investigation. I understand that the animal was not presented for slaughter and did not enter the food chain," she said.
"DARD (Department of Agriculture and Rural Development) has in place robust and rigorous control and surveillance measures around BSE."
A suspected case of "mad cow disease" has been identified in the Republic of Ireland, the Irish government has said.