Gardaí are aware of threats being made at a recent rally against the officer who shot dead George Nkencho and are to seek to speak to the man who made the comments.
In addition, An Garda Síochána is "very concerned" by "lies" being circulated widely online by "fascists and racists" falsely stating that the dead man, who had mental health problems, had more than 30 criminal convictions.
In fact the 27-year-old had no criminal convictions at all.
"We are aware of and are closely monitoring social media. We have concerns about escalating racial tensions," said a well-placed security source.
"On the one hand, there were threats being made against the garda who fired the shots. On the other side, lies are being circulated about the man who died claiming he was a criminal when he was not.
"Unfortunately these lies went viral and lots of right-minded people in society think the dead man was a criminal when he was not. All of these things stoke the fires of racial tension."
Mr Nkencho was shot in the garden of his home in west Dublin at 12.35pm last Wednesday and was later pronounced dead in hospital.
Earlier he had twice been tasered by gardaí, as well as pepper sprayed, after he attacked a Eurospar manager and threatened gardaí with a knife during a 30-minute stand-off during which he was repeatedly asked to drop the knife he was brandishing.
The Garda Síochána Ombudsman Commission (GSOC) is investigating the shooting at Manorfields Drive in Clonee, on the Dublin-Meath border.
Security sources say they are "aware" of a video of a man at a rally organised in the aftermath of the shooting making threats against the garda who used the lethal force.
But sources stressed the situation is "sensitive" and "gardaí are aware that emotions are running high and that the family and friends of the man who died are of course very upset".
GSOC will examine whether gardaí have any criminal case to answer or whether it was justifiable use of force in shooting Mr Nkencho. Five shots were discharged by one of two officers deployed from the Armed Support Unit to the scene.
The garda ombudsman will also examine whether officers followed correct policies and procedures in the run-up to the fatal shooting.
Two well-placed security sources told this newspaper they believe An Garda Síochána "acted appropriately" in the incident.
Since the tragedy, protests have been held outside Blanchardstown garda station and elsewhere.