A 60-second clip, titled "Edinburgh sleeping police", appears to have been captured by a member of the public, known as calumbro.
In a running commentary, the man claims it is 5am and he has just found the policemen at the A720 Edinburgh bypass at the Lothianburn junction.
Oblivious to the amateur cameraman taking close-ups of their faces, the two officers are described as "snoozing and dribbling" during the video.
Lothian and Borders Police confirmed that the men in the clip are their officers and said an investigation had been launched.
A spokesman for the force said: "Lothian and Borders Police are committed to ensuring all of our officers and staff perform their duties to the highest standard.
"We are aware of the online video clip, and an internal investigation is already under way to determine the circumstances surrounding the incident."
During the clip the cameraman says: "Okay, this is, eh, about five o'clock in the morning. I'm up next to the bypass. There's two policemen sleeping in their car."
He also takes footage of the patrol car's number plate.
He signs off with: "Two coppers sleeping."