The pilot was forced to take evasive action during the incident which happened at about 13:10 on Friday.
The unmanned craft was flying about 20-30 metres away from a Loganair flight from Shetland at about 4,000ft.
No-one was injured and the plane successfully landed. Police believe the drone may have been piloted from the Bathgate or Armadale area.
The force is reminding all drone users to operate their crafts within the guidelines set out by the Civil Aviation Authority.
Ch Insp Barry Blair, of Police Scotland, said: "The pilot was forced to take evasive action to avoid a collision whilst preparing to land the plane. Had these aircraft collided the consequences could have been far more serious.
"We are working alongside the aviation authorities to identify the pilot of the drone and determine the full circumstances surrounding this incident.
"Drone users are reminded only to operate their crafts within the guidelines set out by the Civil Aviation Authority and should consult the legislation should they have any queries about appropriate drone use."
A spokesman for Edinburgh Airport said: "People should be in no doubt of the seriousness of this issue.  We would ask that all users understand the legislation and guidelines surrounding drone usage and behave responsibly."
A drone has been involved in a near-miss with a plane making a descent into Edinburgh Airport.