A tourist photographed the bird of prey last week as it was flying over the Aberdeenshire village of Crathie, close to the royal family's Balmoral estate in the Cairngorms National Park.
The picture showed the eagle hovering with the trap clamped around its talons and a chain hanging from it.
The incident is being investigated by the Royal Society for the Protection of Birds (RSPB).
RSPB Scotland's head of investigations Ian Thomson told The Independent: "There is no way a bird of prey could become caught in a legally set trap and as such it is absolutely clear this incident is a result of criminality."
The same style of traps have been regularly used in illegal trapping activities to catch birds of prey on grouse moors.
Golden eagles, which can grow to have a wingspan of nearly 90cm, are the top predator in Scotland and protected under the Wildlife and Countryside Act 1981.