They can be seen with their upper bodies hanging out of the windows of the moving squad car.
The 05 vehicle seemingly has its hazard lights on.
Investigators will be able to check the car's logbook to see who was driving the car and who was their "observer" and passengers.
There were some suggestions the picture is a fake.
Users questioned how the driver could be driving the car while 'surfing' at the same time.
The Garda Press office has confirmed that an investigation into the shocking picture is continuing.
A Garda spokesperson said: "We are aware of the image circulating on social media sites and we are enquiring into all the circumstances of the photograph."
If the image is deemed real, the officers involved are sure to face serious disciplinary action.
The 'surfing' activity was condemned by the Road Safety Authority (RSA) as "breathtakingly stupid".
And 'Liveline' was flooded with callers demanding action if the photograph is found to be authentic.
One caller told Joe Duffy: "These people need to be dealt with very seriously as this is the thing the National Roads Authority (NRA) is talking about.
"If the gardai saw young lads doing this they would have them up before the courts. You need to be lead by example."