Club security staff became concerned about the light aircraft, which made a "sharp turn" and flew "erratically" while the team was playing Brighton.
The club and The Civil Aviation Authority (CAA) hope supporters will have captured footage of the plane.
It is alleged the plane breached CAA guidelines by flying so low.
A club spokesman said: "The context of this appeal is that this plane was spotted flying erratically over the ground a few days after the Paris bombings."
He added: "This caused a potentially serious safety issue at the game and, as part of the investigation, the CAA is seeking video or still photographic evidence from anyone who may have witnessed the incident.
"The CAA said that if they have footage or stills they should be able to calculate the plane's height."
Football fans have been asked to share footage of a plane which flew at low altitude over Burnley's stadium nine days after November's Paris bombings.