Labour Senator Lorraine Higgins, who failed to win a seat in the European Parliament, has said gardaí are investigating a death threat she received last week.
Ms Higgins said she had been subjected to horrendous abuse on Facebook and Twitter in recent weeks following an appearance on RTÉ's Prime Time.
Speaking on RTÉ's Today with Sean O'Rourke, she said the individual involved said she would be "hatcheted in the face".
Meanwhile, Dublin MEP Lynn Boylan said she would condemn anyone who would make death threats to politicians and people in the public eye.
She said that social media is a nasty and unpleasant place, where people find it easy to insult others, but that this is an unfortunate reality.
The Sinn Féin MEP also said she would serve a full five-year term.
Ms Boylan was the first candidate declared elected in Dublin following the third count, after securing more than 83,000 first preference votes.